If the cars are parked in a few secondary lanes, then the city is subsidizing the parking.
Otherwise there will be some sort of commercial parking facility.
In any case, if I go on the Great American Road Trip, I am going to have a single vehicle the whole time, yes? (Assume I rented it for a month, and that it's fully autonomous in a city.)
I don't want to load and unload all of the luggage, including the tent and stove, when I visit a city. So where do I park it?
If I rented it in Florida, and park it in L.A., does the car owning company in Florida own parking spots in L.A.? Or does it have a parking agreement with the parking facility in L.A.? Or do I contract directly with a third party parking company?
My guess is there will still be third party parking services. With the option of renting a space that's cheap but 15 miles away (adding $5 in fuel and a 25 minute latency) or a local place that's more expensive, but where I can easily fetch the suitcase I forgot in the car.
So you normally don't take a taxi out of the city, and it is such a niche activity that I'd assume that it would be a different kind of speciality business with much higher costs that allow it to solve these problems. E.g. Private parking lots like those today.
Most companies, including large national ones, don't own the buildings that their stores / offices are in. They don't want to get into the real estate business and would rather someone else handle it.
The taxis in Beijing have their little hang outs where they park at 4am in the morning. Parking is notoriously hard in our city, but at 4am, the roads are deserted and they spill into a few second lanes without the police caring (and they are all gone by 6 as peak time is starting). Not a problem compared to the wild parking habits of private personal cars.