The following two sets of parents are alleged to be those of our subject.

Father said to be:  Robert HERRINGTON of Suffolk
Mother said to be:  Joan JENTILMAN

Father said to be:  James HARINGTON of Somerset
Mother said to be:  Ann CLINTON

Of the above, the former is at least plausible (Essex is adjacent to Suffolk, while distant from Somerset), though neither is proven.  The latter has no basis in fact and has obviously been claimed because of James's illustrious ancestry.  The bottom line is that Robert's parentage has not been proven on paper.  The only solid evidence we have, at this point, is that he is a DNA match with a HARRINGTON from Essex, and he's not a match with descendants of Benjamin HARRINGTON of RI.

1.  Anon.  1894-1906.  Watertown Records comprising the First and Second Books or Town Proceedings with the Lands Grants and Possessions also the Proprietors' Book and the First Book and Supplement of Birth Deaths and Marriages.  4 vols. Historical Society of Watertown, Watertown, MA (online at AmericanAncestors.org as "Vital Records of Watertown, 1630-1825").

Vol. 1, First Book
p. 14	1649... Suƒann Herington : Daughter of Robert & Susan Herington : Bornn the 18 of August... Robert Herington & Suƒan George Maryed the firƒt of october...
	Note:  these are the first/earliest Herington/etc. entries in the book.
p. 16	1651... John Herington Sonn of Robert & Suƒan Herington Borne the 24 of Auguƒt
p. 17	1653... Robert Herington sonnn of Robert & Suƒan Herington borne the Laƒt of the 6th month
p. 18	1655.. George Herington, the Sonn of Robert & Suƒan Herington borne the 24 of November
p. 19	1657... Danill Hearington sonn of Robert & Suƒann Hearington borne the firƒt of November
p. 22	1659... Joseph Herrington sonn of Robert & Suƒan Herrington born the 28 of December
p. 24	1661... Benjamin Herington son of Robert & Suƒannah Herington Borne 26 : 11 m
p. 25	1663... Mary Herington daughter of Robert & Suƒanna Herington Borne the 12 of January
p. 27	1665... Thomas Herrington son of Robert & Suƒannah Herington Borne 20 2d mo
p. 28	1666... Samuell Heringtton son of Robertt & Suƒann Herington Borne the 18 of December
p. 30	1668... Edward Harington son of Robert & Suƒan Borne the 2d of March
p. 34	1671... Sary Herington daughter of Robert and Suƒan Harrington borne 10 of May
p. 36	1673... David Harington son of Robert and Suƒan Harington borne firƒt of June
p. 39	1675... David Herrington son of Robert and Suƒan Herrington dyed the 11 of march
p. -	-
Vol. 2, Second Book
p. 1	Deaths... Susana Herington wife of Robert Herington Deceased the 6th of July 1694

2.  Clarence Almon Torry.  1985.  New England Marriages Prior to 1700.  New England Historic Genealogical Society, Boston, MA (reprinted 1997, Genealogical Publ. Co., Baltimore, MD).  As a compilation of published sources, not original documents, Torrey should be considered a secondary source, not a primary one.

p. 345

HERINGTON, Robert (1616-1707) & Susan/?Susanna GEORGE (ca 1632-1694); 1 Oct 1649, prob 1648; Watertown

3.  Census Index:  Colonial America, 1607-1789 (Broderbund CD-310):

1652 Fidelity Oath	Harrington, Robert	MA	Middlesex Co.	p. 187
1663 Freeman List	Harrington, Robert	MA	Massachusetts Colony

4.  Frederick A. Virkus, ed.  1964.  Immigrant Ancestors: a List of 2,500 Immigrants to America before 1750.  Genealogical Publ. Co., Baltimore, MD (Broderbund CD-354):

1634

Harrington, Robert

Watertown, MA

p. 36

5.  John A. Farmer.  1829.  A Genealogical Register of the First Settlers of New-England…  Carter, Andrews, & Co., Lancaster, MA (repub. 1964/83 by Genealogical Publ. Co., Baltimore, MD; Broderbund CD-354):

1642

Harrington, Robert

Watertown, MA

p. 135

6.  Patricia Ann Scherzinger.  1996.  Colonial Americans of Royal and Noble Descent:  Alleged, Proven, and Disproven.  Heritage Books, Bowie, MD.

p. 248	HARRINGTON...   John - b 1651, m Hannah Winter - desc/o Robert Harrington LDS82
p. 249	HARRINGTON (cont.)   Patience - chr 1697, m David Stowell - desc/o Robert Harrington LDS82   Robert - b 1616, m Susannah George - the immigrant LDS17; royalty not proven   Sarah - b 1671, m Joseph Winship - desc/o Robert Harrington LDS17   Susanna - b 1688, m Joshua Kendall - desc/o Robert Harrington LDS85   Thomas - b. 1665, m Rebecca Bemis - desc/o Robert Harrington LDS85
p. 562	LDS = Michel L. Call, Royal Ancestors of Some LDS Families, 1972
There's no way to tell, from this source, where the unproven links lie, but the shakiest link is probably between the immigrant and his alleged parents in England.

7.  Fredrick Lewis Weis.  1958.  Early Generations of the Family of Robert Harrington of Watertown, Massachusetts, 1634, and Some of His Descendants.  Self-published, Worcester, MA (online at HeritageQuest at Genealogy.com):

I'm quoting rather extensively from Weis because I think he erred with his speculations as to the parentage of our subject, and I want to be clear as to just what he said.  Unfortunately, because Weis is a highly respected researcher and because this work is in print, people have "run with them" as if they were proven.  I implore everyone to resist accepting any parents for our subject without solid documentation, especially as Y-DNA testing has connected two patrilineal descendants of Robert with the HARRINGTONs of Essex.  The Essex HARRINGTONs may yet genetically connect with the Suffolk HARRINGTONs, but unless and until they do, we should refrain from making the connection.  A bad connection is not better than no connection.
p. 7	Harrington The best clue to the parentage of Robert Harrington of Watertown, Massachusetts, is that he came from Southwold, co. Suffolk, England.  His father, if our surmise is correct, died before 1622 and his mother died in 1630, but he probably lived with his mother and step-father, the Reverend John Younges, between 1622 and 1630, and perhaps with his step-father until 1634.  By 1634, he would be 18 years of age (he died at Watertown, Massachusetts, in 1707, aged 91 years, thus he would have been born in the year 1616), and in those days at 18 he would have been considered able to support himself.  At that time, Deacon Thomas Hastings, perhaps a kinsman, at all events a friend, appears to have paid his passage to America, on condition that Robert would be his helper for six years. We know that Robert and the family of Deacon Hastings embarked in the ship "Elizabeth," William Andrews, Master, at Ipswich, co. Suffolk, on 10 April 1634, with many other Suffolk county families.
Weis's opening sentence is without support.  We do not know that Robert came from Suffolk.  There is no Robert HARRINGTON on the ship's list of the Elizabeth (see following sources).
p. 7 cont'd	Also we know that Watertown was settled by pioneers nearly all of whom came from Suffolk and Essex counties in England.  Bond's History of Watertown names more than forty prominent Watertown families who came from Suffolk county alone.  Furthermore, when Robert's time was up in 1640, Mr. Hastings gave him a homestall in Watertown. Fortunately, we know more than this.  A Robert Herrington was baptized at Southwold, co. Suffolk, 1 October 1616, son of Robert Herrington, who had married on 7 March 1613, Joan Jentilman, baptized 5 July 1603, the daughter of William and Agnes Jentilman.  (Doubtless she was several years old at the time of her baptism.)  This Robert Herrington was the son of another Robert Herrington, who had been bailiff of Southwold for many years.  Joan (Jentilman) Herrington marrried, second, 25 July 1622, the Reverend John Younges.  He was "John: Jonges: of St. Margretts: Suff: Minister: aged 35 years" who tried to leave for New England in May 1637, but was forbidden by the authorities to do so...
p. 8	While we have not seen positive proof that Robert Harrington of Watertown, Massachusetts, is identical with the Robert Herrington who was baptized at Southwold, co. Suffolk, England, it is highly probable that he was...
One wishes Weis had not speculated here because subsequent researchers have accepted this scenario unquestioningly.  It is premature to be making this connections.

7.  Peter Wilson Coldham.  1987-c1993.  The Complete Book of Emigrants.  4 vols.  Genealogical Publ. Co., Baltimore, MD (Broderbund CD-350):

Vol. I, p. 31

30 April 1634.  Passengers from Ipswich embarked in the Elizabeth of Ipswich, Mr. William Andrews, for New England:  John Sherman 20; Joseph Mosse 24; Richard Woodward 45 and Rose his wife 50, George Woodward 13, John Woodward 13; Edmond Lewis 33 and Mary his wife 32, John Lewis 3, Thomas Lewis 9 months; John Spring 45 and Elinor his wife 46, Henry Spring 6, John Spring 4, William Spring 9 months; Thurston Raynor 40 and Elizabeth his wife 36, Thurston Rayner 13, Joseph Raynor 11, Elizabeth Raynor 9, Sarah Raynor 7, Lidia Raynor 1, Edward Raynor 10, Elizabeth Kemball 13; Thomas Skott/Scott 40 and Elizabeth his wife 40, Elizabeth Scott 9, Abigail Scott 7, Thomas Scott 6; Henery Kemball 44 and Susan his wife 35; Richard Kemball 39 and Ursula his wife, Henry Kemball 15, Richard Kemball 11, Mary Kemball 9, Martha Kemball 5, John Kemball 3, Thomas Kemball 1; John Lavericke 15; Isaacke Mixer 31 and Sarah his wife 33, Isaack Mixer 4; Martha Scott 60; George Munnings 37 and Elizabeth his wife 41, Elizabeth Munnings 12, Abigail Munnings 7; John Bernard 30 and Phebe his wife 27, John Bernard 2, Samuell Bernard 1, Thomas King 15; Thomas Kilborne 24 and Elizabeth his wife 20; John Crosse 50 and Anne his wife 38; Robert Sherin 32; Humphry Bradstreet 40 and Bridgett his wife 30, Anna Bradstreet 9, John Bradstreet 3, Martha Bradstreet 2, Mary Bradstreet 1; Henery Glover 24; William Blomfield 30 and Sarah his wife 25, Sarah Blomfield 1; Robert Day 30 and Mary his wife 28; Sarah Reynolds 20; Robert Goodall 30 and Katherin his wife 28, Mary Goodale 4, Abraham Goodale 2, Isaacke Goodale 1/2; Samuell Smithe 32 and Elizabeth his wife 32, Samuell Smith 9, Mary Smith 4, Elizabeth Smith 7, Phillip Smith 1; Thomas Hastings 29 and Susan his wife 34; Susan Munson 25; Martin Underwood 38 and Martha his wife 31; Henery Gouldson 43 and Anne his wife 45, Anne Gouldston 18, Mary Gouldston 15; William Cutting 26; John Palmer 24; Danyell Pierce 23; John Clearke 22; John Firmin 46; Rebecca Isaacke 36; Anne Dorifall 24. (PRO:CO1/8/pp.102-103).

I quote the list in full to demonstrate that not only were adult males listed (as is the usual case), but that women and children (down to infants) were also listed.  If Robert HARRINGTON had been on this ship, his name surely would be listed, but it's not.  (Susan MUNSON was the HASTING's paid servant.)  I checked several other transcriptions (see below), and none of them included our subject, either.

8.  Passengers of the Elizabeth, Master William Andrewes, April 1634 (online at the Winthrop Society web site [link died]).

9.  The Elizabeth (online at Scott Billigmeier's "Hastings Memorial" web site [link died]).

10.  Frank R. Holmes.  1923.  Directory of the Ancestral Heads of New England Families, 1620-1700. [?], NY (repub. 1984 by Genealogical Publ. Co., Baltimore, MD; on Broderbund CD-504):

p. 109

HARRINGTON Place name from parish of Harrington, County of Cumberland, Eng., corrupted from Haverington so called from haver, Dutch, haber, Teutonic, oats, ing a field, and ton.  The town in or surrounded by oat fields. EDWARD, resident of Charlestown, Mass., 1643. RICHARD, freeman at Charlestown, Mass., 1647. ROBERT, b. Eng., 1616, settled at Watertown, Mass., 1634, proprietor 1642. SAMUEL, at hatfield, Mass., 1679.

11.  Henry Whittemore.  1967.  Genealogical Guide to the Early Settlers of America… Excerpted and Reprinted from "The Spirit of '76" Volumes V-XII, September, 1898 - May, 1906.  Genealogical Publ Co., Baltimore, MD (Broderbund CD-504):

p. 36	BEMIS. JOHN BEMIS, of Watertown, Mass., married Mary, daughter of Robert Harrington, and had Beriah 1681, Susanna, Joseph 1684, John, Mary, Samuel, Lydia, Hannah, Isaac, Abraham, 1703, Susanna and Hannah, twins, 1705. He resided in that part which became Waltham.
p. 238	ROBERT HARRINGTON, of Watertown, 1642, married 1648, Susanna, daughter of John George, had Susanna, born 1649; John 1651; Robert, 1653; George, 1655, killed by Indians at Lancaster, 1676; Daniel, 1657; Joseph, 1659; Benjamin, 1662; Mary, 1664; Thomas, 1665; Samuel, 1666; Edward, 1669; Sarah, 1671; and David, 1673, died young; and the father died 1689. His widow married Eleazar Beers the next year, and in 1705, Peter Clayes.
It wasn't Robert who died in 1689, it was John CUTTING, husband of his daughter, Susanna (HARRINGTON) CUTTING, and John died in 1669, not 1689.  It wasn't Robert's wife, Susanna (GEORGE) HARRINGTON, who remarried to Eleazar BEERS and Peter CLAYES/CLOYES, it was Robert's daughter, Susanna (HARRINGTON) CUTTING BEERS CLOYES, who married three times.  Robert HARRINGTON died in 1707, outliving his wife, Susanna (GEORGE) HARRINGTON, who died in 1794.

12.  Charles Henry Pope.  1900.  The Pioneers of Massachusetts.  C.H. Pope, Boston, MA (republ. 1965/98 by Genealogy Publ. Co., Baltimore, MD; Broderbund CD-504):

p. 214

HARRINGTON, ARRINGTON, HERRINGTON, ERRINGTON... Robert, Watertown, propr. 1642; took oath of fidelity in 1652; frm. May 27, 1663. Town officer; mill-owner. He m. Oct. 1, [1647,] Susan George; ch. Susan b. Aug. 18, 1649; John b. Aug. 24, 1651; Robert b. 31(6) 1653; George b. Nov. 24, 1655; Daniel b. Nov. 1, 1657; Joseph b. dec. 28, 1659. The wife d. July 6, 1694. He d. May 11, 1707, ae. 91.  Will names sons John, Daniel, Benjamin, Samuel, Thomas, Edward; daus. Susanna Beers, Mary Bemis, Sarah Winship; Joanna Ward, late wife of his son Joseph, and her son Joseph.

13.  Frederick Adams Virkus, ed.  1942.  The Compendium of American Genealogy: the Standard Genealogical Encyclopedia of the First Families of America.  Vol. VII.  The Institute of American Genealogy, Chicago, IL (repub. 1968 by Genealogical Publ. Co., Baltimore, MD; Broderbund CD-200):

p. 856

Immigrant Ancestors (Biographies)... HARRINGTON, Robert (1616-1707), from Eng. in the "Elizabeth," 1634; owned property at Watertown, Mass., 1642-44; freeman, 1663; selectman, 1697, 1681-83, 85, 1691-1709 (sic); m 1649, Susanna George (1632-94).

There were numerous lineages containing Robert, which I did not extract, but a quick scan shows that many contain serious mistakes, and some are bogus due to an attempt to connect Robert to royal ancestors in England.  These are patron-submitted pedigrees and, as such, are as unreliable as the LDS Ancestral File, WorldConnect, or Broderbund WorldFamilyTrees.  It is well known that Virkus is an unreliable source, so CAVEAT EMPTOR!

14.  Anon. 1975. Founders and Patriots of America Index.  National Society of Daughters of Founders and Patriots of America, Washington, DC (repub. 1989/93 by Genealogical Publ. Co., Baltimore, MD; on Broderbund CD-506; italics indicate a founder, non-italics indicate a patriot):

p. 103

HARRINGTON... Antipas (Sept. 30, 1753-Jan. 27, 1803) m. July 14, 1774 Levinah Brigham XXX, 177; XXXII, 65... John (Feb. 28, 1742-....) m. Dec. 3, 1760 Mary Wooten XXVI, 206, 207 Peter (May 4, 1751-July 29, 1813) m. Feb. 9, 1775 Anna Hammond XIV, 69 Robert (1616-May 11, 1707) m. Oct. 1 1649 Susanna George MASS XIV, 69; XXVI, 206, 207; XXX, 177; XXXII, 65

Antipas⁶ [John⁵, John⁴, George³, Thomas², Robert¹]
John⁵ [Henry⁴, John³, John², Robert¹]
Peter⁴ [Nathaniel³, Edward², Robert¹]

15.  Francis G. Jenkins.  1991.  "Benjamin Howland, a Soule Descendant."  The Mayflower Descendant 41(2): 125-130.  FWIW…

Thankful5 HARRINGTON [George4, George3, John2, Robert1] married Benjamin6 HOWLAND [Samuel5, Samuel4, Nicholas4, Zoeth2, Henry1]. The wife of Samuel4 HOWLAND (paternal grandmother of Benjamin6) was Sarah4 SOULE [William3, George2, George1]. George1 SOULE was a passenger on the Mayflower. In other words, Thankful (HARRINGTON) HOWLAND's descendants are Mayflower descendants, though she is not.

16.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).