PARENT is a French name, common in French Canada and bordering states in the U.S., plus Louisiana.  The double-R is a later addition.

— born in New Orleans, Orleans Co., LA:
6.  Frank M. PARRENT, b. Aug 1857
7.  T. E. PARRENT [male], b. 1860/1

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

Chancey PARRENT Spouse: Rowena CROCKER Marriage: 22 Oct 1839, Monroe, Mississippi Source: "Extracted marriage record for locality listed in the record."

1b.  LDS.  Family Search: Internet Genealogy Service:  AF - Ancestral File (online at FamilySearch.org).

Chauncey PARENT Birth: Abt. 1816 Marriage: 22 Oct 1839, Monroe Co., MS Rowena CROCKER Birth: 4 Sep 1820, Laurens Co., South Carolina Father: James CROCKER (1784-1830); Mother: Susannah NISWANGER (1795-1852) Source: patron submission

2.  1840 Census Index/Images (online at Ancestry.com, Image #23-24 of 59 — indexed "Channey Paerant"; extracted by Diana Gale Matthiesen):

1840

MS

Monroe Co.

Roll 218

pp. 69B-70A

Ln. 7

Chauncy Parrent

000 020 - 000 100

000000-110000

0002000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
2 white males	20-29	1810-1820	= ? = Chauncey (b. 1813/4)
1 white female	15-19	1820-1825	= Rowena (b. 1818/9)
1 female slave	9 or under	1830-1840	= ?
1 female slave	10-23	1816-1830	= ?
2	persons employed in manufacture and trades

3a.  1850 Census Every-Name-Index/Images (online at Ancestry.com, Image #79 of 79):  City of New Orleans (Ward 2, Municipality 1), Orleans Parish, LA, Roll, p. 84B, 1490/1500, enumerated 4 Sep 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

C Parent	36	M		Carpenter		NY
Rozina "	31	F				SC
James  "	10	M				Miss
Everet "	8	M				"
Charles "	6	M				"
Orlando "	4	M				"
Marguerite	6/12	F				"

3b.  1850 Census Every-Name-Index/Images (online at Ancestry.com, Image #93 of 103):  Western Division, Monroe Co., MS, Roll 378 (Book 1), p. 109B/218, 667/667, enumerated 17 Dec 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

Roena Parent	28	F				SC
James Parent	11	M				Miss
Everett Parent	9	M				"
Charles Parent	8	M				"
Orlander Parent	6	M				"
Susan Parent	9/12	F				"

I would have assumed Chauncey passed away between their enumeration in New Orleans on September 4th and Rowena's enumeration in here in Mississippi, except that Rowena had two more children after Susan.  Listed next to Rowena's remarried mother, Susannah (NISWANGER) MONCRIEF.

4.  1860 Census Every-Name-Index/Images (online at Ancestry.com):  can't find, and I really looked.  I can't find Rowena's brother, James CROCKER, either.
 

WARNING:  There is something very wrong with the 1870 census of Hardeman Co., TN, over and above the sloppy handwriting and the omission of given names.  The pages are out of order, not just the stamped page numbers, but the handwritten Page Nos. as well.  Plus, the visitation numbers have been written over, possibly in someone's misguided attempt to make the pages appear to "fit."  So, be cautious with any situation where you have a family bridging two pages.  Make certain putting the pages together makes sense and, if not, scan the pages for the one that does.  Below, I've pieced these two pages together based on what is known about the families.

5.  1870 Census Every-Name-Index/Images (online at Ancestry.com):  Civil Dist No. 11, Hardeman Co., TN, Roll M593_1533, enumerated 3 Aug 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

1870:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).

Image 25 of 40 Page No. 25 p. 255A

1

2

3

4

5

6

7

8

9

10

15

16

17

19

171

172

Crocker J W

41

M

W

Merchant

25,000

5,000

S Carolina

/

/

/

"    M A

31

F

W

Mississippi

/

/

"    M W

11

F

W

Tennessee

/

/

/

"    J M

9

M

W

"

/

/

/

Image 40 of 40 Page No. 37 p. ???B

247

247

Crocker H S

6

F

W

Miss

"    W M

4

M

W

Tenn

"    C C

2

M

W

Miss

McFarlane J C

21

M

W

"

/

/

/

/

Parrent R

49

F

W

S Carolina

/

/

"    Susan

16

F

W

La

/

/

/

"    F M

13

M

W

"

/

/

/

"    T E

9

M

W

"

/

/

/

"    R

76

F

B

S Carolina

/

/

James W. CROCKER is Rowena's brother.  As Rowena is not the head-of-household, it's probable she was not the informant, hence the discrepancy in Susan's age.

6.  1880 Census Every-Name-Index/Images (online at Ancestry.com):  can't find.

7.  1890 Census:  the 1890 Census Population Schedules were destroyed.

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