Is it permissible for my father to benefit from the profit generated by selling the house that was previously mortgaged with usury, in order to pay off other debts or for investment?

Mortgaging in itself is not a sin; rather, it is a documentation of debt. The sin arises if the mortgage is for an interest-based transaction. The increase in the house's price over time is permissible and there is no sin in it. It is obligatory for the parents to repent from previous usurious transactions and to rid themselves of their effects, as Allah Almighty says: "Those who consume interest will not stand [on the Day of Resurrection] except as one stands who is being beaten by Satan into insanity. That is because they say, 'Trade is [just] like interest.' But Allah has permitted trade and forbidden interest. So whoever has received an admonition from his Lord and desists may have what is past, and his affair rests with Allah. But whoever returns [to interest] - those are the companions of the Fire; they will abide eternally therein." Al-Baqarah/275.