I don't know about the parking but I have very bad arthritis. The first time we went to Disney I tried walking and had a miserable time. A woman came up to me, she could tell I was in pain, and told me to stop being vain and rent a wheelchair. The next day I did and it was okay but not for me. Having my husband push me around made me a nervous wreck, he pushed me into someone once, and you are dependant on someone. So the next day, I rented an electric chair. It does cost more ($30) but is worth it for me. I'm not depending on anyone and if they go on a ride I don't want to I can go off on my own. Maybe your father would perfer this. There are alot of seniors that rent the chairs.
As for the parking, I don't know about dropping off someone but I have handicap parking and I found that usually I had to walk further with the handicap parking than if we just parked in the regular spot and took the tram. We never take the handicap parking now.
I hope you can talk him into renting a chair. He'll enjoy the parks alot more.