The story passed around online is that Elijah Peters was the illegitimate son of Sarah "Sally" Peters, possibly by a man named "Carl Gabbard". I have never been able to find any information about this mysterious Carl Gabbard person.
About a month and a half after Elijah's birth his mother married Matthias Gabbard who would have raised Elijah as his own. I wouldn't be surprised if Matthias was his actual father, but perhaps not since he didn't take the Gabbard name.
Mathias Gabbard and Elijah Peters both appear in the Clay Co., Kentucky in the 1840 Census, but Elijah was already an adult starting his own family at that time.
The 1850, 1860 & 1870 Censuses show both families moved to Owsley Co., Kentucky. Contrary to the estimated death years some give for Mathias and Sally, I think they probably died during the 1870s as I haven't found them in the 1880 Census. Elijah, however, can be found moved to to Jackson Co., Kentucky, where he died soon after.
The 1830 Census also shows Mathias Gabbard in Clay Co., Kentucky with a son in the right age range to be Elijah, though it only shows 2 daughters born so far, whereas the information I have seen for Mathias & Sarah shows them with 3 daughters at that time (Catherine "Kitty", Matilda & Margaret).
The 1820 Census shows Mathias Gabbard in Mercer Co., Kentucky again with a son in the right age range to be Elijah and only 1 daughter at that time.
I'm descended from Elijah's son Samuel Squire Peters.