You could also greatly increase parking density in parkcades if all vehicles had autonomous parking. For example, it would be no problem to box cars in if vehicles could move on their own to let them out. I bet you could easily double parking density.
Taken to the limit of the idea, the whole parking structure could be entirely sealed to the outside, and maybe even evacuated or filled with a non-oxygenated medium (e.g. nitrogen) to prevent car fires and reduce rusting from long-term storage. Sort of like a modern grain silo.
I'm somewhat uncomfortable with parking infrastructure that assumes that all the cars are working and "gassed up." If the car boxing mine in is left to sit in the parking garage for a few months (as might happen in, say, an airport's long-term parking lot), its battery will likely go flat and it'll be incapable of getting out of my way.
Even having automatic standardized charger coupling for every parking space wouldn't really work, since the car boxing mine in could have just plain rusted.
Current infrastructure doesn't have this problem: if the car beside mine can't get out of the garage on its own, that's between the garage and a tow truck, but is no problem of mine.
There is also the alternative that people will move further away from their work, because they can use the commute time -- essentially then, they would leave their homes at the same time, start working and then leave their offices so that they are home again at the same time, which means (assuming everybody lives at different distances from their workplace) that you automatically get a stackering effect.
Autonomous fleets can also be parked in towers, because a robot car doesn't care if it takes 10 minutes to get out of a tower, as long as the throughput is high enough.
Eventually you will need heavy rail to increase throughput for dense enough places, but that's still a separate problem today.
That said, only a percentage of people would want to just discard the car they've spent considerable money on.
Self-driving fares can never be as cheap as regular driving seems. With regular driving, you are silently depreciating the value of your vehicle whereas a self-driving car would have to include depreciation in each ride.
However, for people who do (or plan to) use bicycles or mass transit a significant proportion of the time, being able to fall back on an hailed autonomous car in case of rain, haste, whatever is very appealing. It just depends on being able to look at a monthly bill of $200 (or whatever) in on-demand rides and see it as a huge step forward from what you used to pay in all the fixed costs of car ownership whether you used it or not.
The real trick will be having a workable solution for families with kids— that's an area where current taxis/rideshare have a gap, since they're not willing to supply their own carseats for liability reasons.
But when you have that, you double or multiply by N, the road usage. Which is going to produce multiple problems for everyone.
People imagine that once autonomous vehicles reach a certain level, they'll just morph to provide all the benefits of a well-planned public transit system. But I'm doubting common good can just fall out of more chaos.
And hey, the private automobile itself was thrown onto society with effectively no consideration of externalities and brought us to our current heaven/hell of urbanism. So it's quite possible that, despite some talk of planning, autonomous vehicles will just appear, make a hash of things and all the clean-up will happen later, if at all.
The carseat thing is a real missing piece in the puzzle though.
If I own/rent the car, I can just leave them in while I'm doing whatever. If I'm doing Lyft, I'm stuck carrying around baby seats between rides. Which is not appealing.
Of course, these cars occupy space, which means the road they'd be on would be a good deal more occupied. But remember, parking and traffic aren't static per-car, it's all a matter of rates.