I assume it's possible that whatever convictions your father had back in 1978 are not able to be found in the NCIS or the county where the crime was committed. If he can purchase weapons then something isn't showing as it should be. Computers were not common back then and everything was done on paperwork. It's possible it never made it into the system and the county records are sitting in a dusty closet somewhere.
I'm not telling you to have your father lie but if the records can't be found then it's on his conscious on whether he want's to apply and take a chance on getting caught. I've always been one to believe that once someone serves their time they should have all rights restored (All except child molesters and murderers that is) People make mistakes and should get a second chance and sometimes a third. They shouldn't have to pay for them the rest of their life once their penance is paid unless they took a life or took innocence. IMO.
FYI - you can go to the FBI website and pay to have them run a criminal background check on him. The same one they will run when he applies. You can also go to the county or counties where his felonies were commited and have them do the same. If nothing show's up then well, nothing shows up. It's then up to him.