State laws vary considerably, so you should confirm this response with a local elder law attorney. But here are the general rules:
The Medicaid agency will have a claim to recover what it pays out on your father’s behalf from his estate, which would include the house. If he dies before your mother, they must wait to collect on this claim until your mother passes away.
However, if his will gives the house to your mother, then the house will end up in her estate and the state will have the right to recover its costs for her care as well.
So, at the very least your father should execute a will that disinherits your mother. There may be other steps that your father can take to avoid probate and the state’s claim against his house, but these will depend on state practices and on the specifics of your family situation.
So, again, you should consult with a local elder law attorney. Find an attorney near you today for further guidance.