He was probably a grandson of Ailill Flann Bec, and possible cousins were Dáire Cerbba and the famous Crimthann mac Fidaig.
Further confusion is caused by the fact that a certain Láre Fidach is named as the father of Crimthann in one, although not necessarily the oldest, source (Laud 610), because this would rather implausibly make him and Corc brothers.
David Sproule more or less gets around this by arguing that Corc's father should be Lugaid Láre and that the Laud pedigree has been misread and/or is erroneous.
One of the two wives of Conall Corc was Aimend, daughter of Óengus Bolg, king of the Corcu Loígde.
The other, the daughter of the King of the Picts, is stated in the genealogies to have been Mongfind, although she is likely confused with an Irish queen of the same name, who may or may not have been Crimthann mac Fidaig's sister.