Some secondary sources show these parents for Catherine, but I have not only been unable to prove the connection, what I have found so far (viz., the BRUCE bio from The History of Essex County) appears to disprove it:
Father doubtfully:  Benjamin BRUCE
Mother doubtfully:  Philadelphia WHEELER — descendant of Mayflower passengers, Resolved WHITE and his father, William WHITE.
Obviously, the incentive to make this connection is very strong, and even I (dedicated skeptic) am reluctant to let it go.  If anyone has managed to connect Catherine to these parents, despite the bio, please share the proof!  Or if you know who the parents of Catherine really are, please share that, too, so we can give this lapse into wishful thinking a swift burial.

1.  LDS.  Family Search: Internet Genealogy Service:  IGI - International Genealogical Index (online at FamilySearch.org).

Joseph RAZY Birth: 02 Feb 1783, Surry Twp, Cheshire, New Hampshire Father: Joseph RAZY; Mother: Lois Source: Index to Births, Early to 1900. New Hampshire. Registrar of Vital Statistics

2.  1810 Census Index/Images (online at Ancestry.com):  not found, and I did a page-by-page search of Jay, however, there are one or two names lost at the binding edge of each page, so he may have been there, especially as Rufus RAZY and Asahel RAZY were there.

3.  1820 Census Index/Images (online at Ancestry.com; Image #6 of 7 — list was roughly alphabetized; extracted by Diana Gale Matthiesen):¤•

1820

NY

Essex Co.

Jay

Roll M33_69

p. 16B/424

Ln. 18

Rasey Joseph

300010-31010-0-100

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
3 males	9 or under	1810-1820	= Alexander (b. 1817/8) = Oliver (b. 1815) = Otis (b. 1812)
1 male	26-44	1775-1794	= Joseph (b. 1783)
3 females	9 or under	1810-1820	= Eleanor (b. 1819/20) = Zipha (b. 1814) = Delphia (b. 1811)
1 female	10-15	1804-1810	= Mary (b. 1808)
1 female	26-44	1775-1794	= Catharine (b. 1791)
1	person engaged in agriculture

Listed six lines below Asahel RAZEY. 

4.  1830 Census Index/Images (online at Ancestry.com, Image #3 of 4; extracted by Diana Gale Matthiesen):¤•

1830

NY

Franklin Co.

Duane

Roll 90

p. 38

Ln. 9

Racy Joseph

122 000 100 - 011 211

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	4 or under	1825-1830	= Son
2 males	5-9	1820-1825	= Jonah (b. 1824) = Joseph (b. 1822)
2 males	10-14	1815-1820	= Oliver (b. 1815) = Otis (b. 1812)
1 male	40-49	1780-1790	= Joseph (b. 1783)
1 female	5-9	1820-1825	= Daughter
1 female	10-14	1815-1820	= Eleanor (b. 1819/20)
2 females	15-19	1810-1815	= Zipha (b. 1814) = Delphia (b. 1811)
1 female	20-29	1800-1810	= ?
1 female	30-39	1790-1800	= Catharine (b. 1791)

Mary is married, so she can't be the female born 1800-1810.  Our subject is said to be in St. Lawrence Co., but the numbers for the Joseph RACY in St. Lawrence County don't remotely resemble our subject's family, while the numbers for the Joseph RACY in adjacent Franklin County fit our subject perfectly.  (The Joseph RACY in Oneida County is way too young — in his 20's.) 

5.  1840 Census Index/Images (online at Genealogy.com, Image #2-3 of 3; Hastings is mis-indexed at Ancestry.com as being in Berrien County; extracted by Diana Gale Matthiesen):¤•

1840

MI

Barry Co.

Hastings Twp.

Roll 203

pp. 34A-34B

Ln. 6

Joseph Racy

001 210 010 - 000 000 010

0400000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	10-14	1825-1830	= Son ?
2 males	15-19	1820-1825	= Jonah (b. 1824) = Joseph (b. 1822)
1 male	20-29	1810-1820	= Otis? (b. 1812)
1 male	50-59	1780-1790	= Joseph (b. 1783)
0 females	40-49	1790-1800	= Catharine (b. 1791)
1 female	50-59	1780-1790	= ---
4	persons employed in agriculture

Oliver is married, so he's not the male born 1810-20.  Listed one line above married daughter, Eleanor (RASEY) WILKINSON.

6.  1850 Census Every-Name-Index/Images (online at Ancestry.com; Image #4 of 4):  Township of Castleton, Barry Co., MI, Roll M432_346, p. 125B, 918/937, enumerated 16 Sep 1860, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):•

Joseph Rasey	67	M		Farmer		N Hamp
Catharine "	60	F				Mass

Listed next to son, Jonah B. RASEY, who is next to grandson, Nathan BROWN, son of daughter Mary (RASEY) BROWN.

7.  1860 Census Every-Name-Index/Images (online at Ancestry.com, Image #9 of 20 — indexed "Racy"):  Castleton P.O., Castleton Twp., Barry Co., MI, Roll M653_536, p. 853, PN 229, 1834/1713, enumerated 12 Aug 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):¤•

Joseph Rosy	77	M		Farmer		50	N H
Catharine	69	F					Mass

Listed next to son, Jonah RASEY.

8.  Anon.  1997.  Hosmer Cemetery, Castleton Township, Barry County, Michigan.  Barry County Historical/Genealogical Society, Hasting, MI (online at Interment.net; contributed by Sara Colburn):

Rasey, Catherine, 77yr, Nov 15 1863, Row 1/2

Rasey, Joseph, 88yr 5mo 15dy, Jul 18 1869, Row 1/2

They are the only RASEY's listed in the cemetery (no BRUCE or BROWN).

9.  Birthdate calculated using Ben Buckner's Birthdate Calculator.

Name	Deathdate	Age at Death	Result
Joseph RASEY	18 Jul 1869	15d 5m 88y	3 Feb 1781
(Calculated birthdates should always be prefaced with "ca." — you can't be certain the person who calculated their age did it correctly.)

Either the tombstone was mis-read or his age was mis-calculated because we have a civil record giving his birthdate as 2 Feb 1783, which is surely the correct one because his parents were married in 1782.

10.  Anon.  1880.  History of Allegan and Barry Counties, Michigan.  D.W. Ensign & Co., Philadelphia (online at HeritageQuest at Genealogy.com):

p. 419

CASTLETON TOWNSHIP... EARLY SETTLEMENTS... Joseph Rasey and four sons, Oliver, Otis, Joseph H., and Jonah B., residents of St. Lawrence Co., N.Y., removed thence to Ohio in 1836, and to Castleton in 1838.  There they located upon 80 acres on section 13, purchased of Tolman C. Hall, of Battle Creek.  Mr. Clapp's house was opened to them until a log shanty could be erected, to which they then repaired. In Mr. Racey's family was celebrated the first marriage, the parties being Mr. William P. Wilkinson and Miss Eleanor Racey.  Mr. Racey died in the township in 1869.  His son, Jonah B., is now a resident of Nashville; Joseph H. resides in Charlotte, Mich., while Oliver and Otis are both deceased...

p. 421

CASTLETON TOWNSHIP... ORGANIZATION AND OFFICERS... 1843... Oliver Rasey... Highway Commissioners... Oliver Rasey... Justices of the Peace... 1844... Joseph Rasey... Directors of the Poor... Joseph Rasey... Constables... 1845... Joseph Rasey, Overseers of the Poor... Oliver Rasey, Constables... 1846...Oliver Rasey, Justices of the peace... 1847...Joseph Racey, Directors of the Poor... 1848...Joseph Racey... Directors of the Poor... Henry Racey... Constables...

11.  Edward W. Barber.  1897.  The Vermontville Colony.  Robert Smith Printing Co., Lansing, MI (online at Ancestry.com).  On p. 14:

West, in Castleton, just over the town line, a shoemaker, Joseph Rasey, had settled on a wild eighty acres, and to him with a side of sole leather and enough upper leather to shoe the family the boys would go every fall, after the frost had begun to bite, and have a pair of cowhide boots made for winter, going barefoot and enjoying an occasional stonebruise having been the summer custom...

12.  James W. Howell of Baltimore, MD.  Ancestors of James William HOWELL & Sherri Lynn (MARTIN) HOWELL (online at www.jhowell.net).

13.  Erick James Owens.  The Owens and Liedel Family.  FamilyTreeMaker User Home Page (online at familytreemaker.genealogy.com). 

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

15.  Find-a-Grave.