Yeah, sure, we can think of a far future where everything is automated but I think this discussion is more about the near future (near being when self-driving cars are for sale).
The logistics of parking go far beyond getting into the space. How does it pay the parking meter? How does it know if a parking lot is public or not, how does it pay for that? How does it know not to go into the unsafe neighborhood where wheels get stolen?
Compared to the logic involved in actually driving down the road safely these are all easy problems to solve.
But realistically they are problems that won't be had, because in the near future (and probably in the far future as well) most people aren't going to buy their own personal self driving car because not only are they expensive, but it is pointless to have a personal car when it is way more convenient to offload responsibility for the car ownership to someone else.
Instead they will use an app to summon an automated taxi car owned by a corporate entity (Google, Uber, etc), and when they are at their destination that car isn't going to park, it is going to be off to pick up another passenger.
The car owner will have designated secure parking lots for overflow cars that aren't needed out on the roads during non peak hours. The cars would return to that designated lot when not needed.
Sure, I might be overstated the problem of parking, but I still don't see how this is more economical than car ownership. If I have to summon a taxi for everywhere I need to go I need the taxi rates to be a fraction of what they are today.
I can't disagree with you on all of that, but I feel like you might be making the problem harder than it needs to be:
Almost any initial solution is going to be imperfect, and is going to involve mistakes if you try to solve for the entire problem space. Usually, the way to address this is to solve a small subset of the problem. If you don't know how to pay for the parking in a particular place, don't park there. Things like that. You limit the problems that you have to address, and the lessons you learn while applying that technology and the infrastructure that develops around its use alters the difficulty of refining it to address other cases.
I can easily imagine the way that I would solve the parking problem, at least initially: Set it up so that the car can pay wirelessly and make it so that people who own parking spaces can add their carpark to a database as enabled for wireless payment. There are even systems which already exist that do something similar to this; toll roads, toll bridges, et cetera: you put a little device on the front of your car and that records your usage of the relevant infrastructure, which is then charged to your account.
Put a bunch of image recognition for parking signs in the car - good parking space usually follow fairly straightforward formats; I'm not sure whether the signage is defined in statute or regulation, but generally they're much alike. Do some simple correlations on whether the person actually indicates they want to park there to refine the algorithm....
Crime figures are publicly available, so you can hook into that database for avoiding bad areas...
And have a rule that says not to park somewhere if you're unsure.
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In the case of taxis, it's even easier: there are designated taxi ranks where they have to wait. And, considering there commercial enterprise, (there are two types of taxi regulation here, it's a bit weird,) you could rely on the taxi company to fill out a database themselves of other places that the car could wait - it's in their financial interest to do so.
The logistics of parking go far beyond getting into the space. How does it pay the parking meter? How does it know if a parking lot is public or not, how does it pay for that? How does it know not to go into the unsafe neighborhood where wheels get stolen?