Medical Mileage

I drive my (95 year old) father to his medical appointments in my car. He is telling me that he can claim the mileage that I am driving on *his* taxes. This doesn't seem right to me and I don't want to keep track of all that data if I don't have to. I am happy to do it if it is a legitimate deduction for him -- but I need to know what the actual rules are.

 

Thanks!