No Proof of Payoff

Your Dad is certainly the owner of his house, and needs no proof of that.  What he does need is a document (either a satisfaction certificate or a reconveyance deed, whichever is used in your state) to show that the mortgage debt is no longer a lien against the property.

He can look in his county's public records, to see if the lender has placed the document  there.  If so, he has nothing to worry about. It won't matter whether he has a copy or not.   But if the records still show the loan out against his house, then he needs to start making a fuss.  He should both call and write to his mortgage lender, and if nothing happens after a few weeks, write to the head of the mortgage company and also to the state authorities that regulate mortgages.  A public librarian can help him find the right names and addresses.