I think he is the son of Griffin C. and Sarah A. MARKS of Byrd, Goochland, VA.On the 1860 census he is listed as Griffin F. and on the 1870 as Franklin G.
There's a marriage record for Griffin C. MARKS to Sarah A. JOHNSON on 25-Nov-1843 in Goochland Co.Groom's father was Griffin MARKS, Bride's father was Benjamin JOHNSON.
Various censuses put the Griffin C's birth year abt 1824 and Sarah's abt 1825.
There's a user submitted tree at familysearch.org that says Griffin C. MARKS was the son of Griffin MARKS and Susanna WEBB.The elder Griffin was the son of John MARKS, and Susanna the daughter of William WEBB.