Forget parking, the cars would be driving, ferrying people and goods around, as much as possible when not charging. No need to own one, just hail one when you need a ride.
It will be interesting to see how people take to not owning cars. People like customizing their spaces, and I can see people with young children not wanting to deal with all that setup multiple times a day...
That's practical, and for people like you and me who aren't bothered by the idea of not owning our own vehicle, but it's very culturally desirable in the US to own a personal vehicle. Not an unsolvable problem, but it will definitely be a bump in the road.
I do wonder how many people living in cities would still choose to own a car if they could just get a self-driving one on demand.
For example, friends of mine recently bought a summer cottage. It's a little over an hour's drive. They don't have a car, so they currently rent one or try to borrow one when they want to go there. Once there, the car will probably sit still for a week.
Currently, there's no really good solution to this. Renting is expensive if you just need to drive a couple of hours a week. Buying a car might make more economic sense in the long term, but it's still pretty expensive for the purpose. Taxi rides like that cost an arm and a leg too. But if the car could drive itself out of the woods...
Of course it's a question of pricing, and I'm guessing we're still some way off from having self-driving cars navigate dirt paths that pass through people's yards as is the case here.