Next to zero then. You definitely don't need parking companies. Most likely the owners of those vehicles, the commercial companies, would have their own lots for parking in places where it was the cheapest. But third party parking companies with parking lots charging exuberant fees would definitely be a thing of the past. What does a 5x10 piece of land cost of rent for a month at the edge of the city. Next to nothing.

For this to work, I think you are predicting there will only be a handful of large commercial companies in this autonomous future. Otherwise I don't think the economics works out.

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.

Take the taxi model, eliminate the driver, and there it is. There are plenty of cities in the world where taxis are more ubiquitous and owning personal cars are very expensive (parking being a big expense, but also winning the license plate lottery). American centrism doesn't really explain it.

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.

<i>Most likely the owners of those vehicles, the commercial companies, would have their own lots for parking in places where it was the cheapest.</i>

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 amount of surface area needed for cars in motion is strictly greater than the amount of space needed for cars at rest which means traffic lanes can dynamically shift from transit to storage based on variance in demand.