WARNING:  There is not a shred of evidence documenting Richard's parentage, only secondary sources.
Father possibly, but only remotely:  Richard LYON
or
Father probably not:  John LYON
Mother probably not:  Eupham GLEDSTANES
or 
Father definitely not:  Sir Glenn LYON
Mother definitely not:  Lady Priscilla BOWES

1.  Clarence Almon Torrey.  1997.  New England Marriages Prior to 1700.  Genealogical Publ. Co., Baltimore, MD.  On p. 481:

LYON, Richard (-1678) (he calls Mary FITCH "cousin") & 2/wf? Margaret __?__ (-1705) (2nd m ca 1668?, ca 1666); b 1653, b 1649?; Fairfield, CT

The "b 1653, b 1649?" refers to birthyears of earliest known children, as an approximate indicator of marriage date.  Location is deathplace of husband.  Torrey implies a former wife for Richard.  He is the only source I've found, so far, that does so, though most sources have Richard as the first son of Margaret, born ca. 1653.  If the birthdates given by Jacobus (see next source) are close to correct, then either Richard had a former wife or he and Margaret were married earlier than generally believed.

2.  Sidney Elizabeth Lyon, Louise Lyon Johnson, & Albert Brown Lyons, eds.  1907.  Lyon Memorial:  Families of Connecticut and New Jersey, including Records of the Immigrants Richard Lyon of Fairfield and Henry Lyon of Fairfield, with a sketch of "Lyons Farms" by Samuel R. Winans, Jr.  W. Graham Printing Co., Detroit, MI.  Includes a Transcription of the Will of Richard Lyon.

3.  Donald Lines Jacobus. 1932.  History and Genealogy of the Families of Old Fairfield. 2 vols.  Daughters of the American Revolution, Fairfield, CT, p. 393.  Birth dates of the children are from this source.

4.  David Dobson.  1985.  Directory of Scottish Settlers in North America, 1625-1825.  Vol. 5.  Genealogical Publ. Co., Baltimore, MD (Broderbund CD-354).  On p. 145 (boldface added):

Primary Individual: Lyon, Richard; Place: America; Year: 1649. Date of emigration and intended destination or place and date of first mention of residence in the New World.  A majority of the data is (sic) on those who settled in Canada.

"Canada" being mainly Nova Scotia.

5.  Meredith B. Colket, Jr.  1975.  Founders of Early American Families: Emigrants from Europe, 1607-1657.  General Court of the Order of Founders and Patriots of America, Cleveland, OH (Broderbund CD-354).  On p. 185:

Primary Individual: Lyon, Richard; Place: Connecticut; Year: 1649.

6.  John A. Farmer.  1823/1983.  A Genealogical Register of the First Settlers of New-England.  Carter, Andrews, & Co., Lancaster, MA (reprinted 1976/83 by Genealogical Publ. Co., Baltimore, MD; Broderbund CD-354).  On p. 186:

Primary Individual: Lyon, Richard; Place: New England.

7.  Louis F. Middlebrook.  1909.  Register of the Middlebrook Family.  Self-published, Hartford, CT (Broderbund CD-117). On p. 21 (boldface added):

From the Colonial Records of Connecticut, 1665-1677, page 106:  At a Court of Election held at Hartford, May 13, 1669, Presented for freemen from Fayerfield, Mr. Joanes, Mr. Harney, Richd Lyon, John Andrews, Francis Bradley, John Tompkins, Joseph Middlebrook... It is interesting, however, to note that the present home premises of Albert James Middlebrook, of No. 515 Wood Avenue, Bridgeport, Connecticut, are very nearly if not exactly situated upon a portion of the original grant to his ancestor Joseph, who settled in the so-called Pequonnock Fields, about 1649.

Joseph Middlebrook married Mary Batemate, sister of Elizabeth Bateman, wife of Richard's brother, Henry.  The land of William Bateman, father of Mary and Elizabeth, was adjacent to Joseph Middlebrook's, and Richard's land was also in Pequonnock.  It seems possible that Richard and Joseph settled in Pequonnock about the same time, and it would be worth looking into the origins of those who settleed in the area at that time.

8.  Colonial America, 1607-1789 CT Census Index (online at GenealogyLibrary.com):

Lyon, Richard	CT	Fairfield Co.	Fairfield	1671
Lyon, Thomas	CT	Fairfield Co.	Fairfield	1671
Middlebroo (sic), Joseph	CT	Fairfield Co.	Fairfield	1671

9.  William H. Powers.  1921.  Powers-Banks Ancestry.  John Leslie Powers, Ames, Iowa (online at GenealogyLibrary.com), in a section entitled, "The Lyon Connection."

10.   William Richard Cutter.  1913.  New England Families:  Genealogical and Memorial.  Vol. IV. Lewis Historical Publ. Co., New York, NY (reprinted for Clearfield Co. by Genealogical Publ. Co., Baltimore; online at GenealogyLibrary.com).  There is a segment on LYON on pp. 2073-4.

11.  Broderbund.  World Family Trees.