The middle initial F may stand for Fannin. He was listed as Peter Fannin, age 8, on the 1860 census with his mother and stepfather, and as Peter Kidd in 1880. I have not found him in 1870 under either name. (There was, however, a different Peter Fanning in Morgan Co, born 1855, son of William Fanning and Unice Day, who should not be confused. That Peter stayed a Fanning all his life.)

1880 census of Elliott Co., Ky, Devil’s Fork precinct, enumerated 9 June, lists on p. 8 as family #57:
Peter Kidd, 28, farmer; Sallie A., 24, wife; Isaac F., 2, son; Lizzie, 1, dau; Mary, 48, mother; Edmond, 8, bro. All birthplaces Ky Ky Ky.

This is the only Kidd household in Elliott Co. The preceding household is that of Sally’s parents Pleasant Wilson and Nancy Stamper.

1900 census of Elliott Co., Ky, Devils Fork precinct, enumerated June, lists on p. 11B as family #192:
Kidd, Peter, 48, b. Apr 1852, farmer; Sarah, wife, 42, b. June 1857, married 23 years, mother of 11 children, 8 living; Tolliver, Lizzie, dau, 21, b. Jan 1879, married 4 years, no children; John, son, 17, b. Mar 1881; Solomon, son, 17, b. Mar 1883; Ann, dau, 14, b. Nov 1885; Marian, son, 8, b. Aug 1891; Jesse, son, 6, b. Sept 1893; Andrew, son, 4, b. Nov 1895. All born Ky Ky Ky.

Lizzie’s husband William T. Tolliver was recorded in this census year as divorced.

1910 census of Elliott Co., Ky, Devils Fork precinct, enumerated 22 April, lists on p. 4A as family #54/56:
Kidd, Peter, 57, farmer; Sarah A., wife, 52, first marriage for both, married 33 years, mother of 12 children, 8 living; Jesse, son, 16; Jennie, dau, 9. All born Ky Ky Ky.

The following household is headed by their son Saul. The preceding household is that of Sarah’s father and stepmother.

1920 census of Elliott Co., Ky, Devils Fork precinct, enumerated 31 Jan, lists on p. 9B as family #161:
Kidd, Peter, 66, farmer; Sarah, wife, 62. Both born Ky Ky Ky.