My car (a 2010 VW Tiguan) parallel-parks itself - well, in as much as it operates the steering while I handle the pedals and gear changes - current models also reverse into parking bays.
This is also a standard feature on the top-end model of the Ford Focus, and probably a bunch of other cars. They don't even need reversing cameras for it - just some carefully positioned 'parking sensors'.
The biggest advantage to self-driving cars that I see is that I can drive a car into a city center, drop myself off, then have the car wander aimlessly through the streets until I need to be picked up. The cost of the gas is almost certainly cheaper than the cost to park would be.
You don't want it to drive around, that's pointless. (Also, petrol's not cheap everywhere.) It could just zoom off to an out of/edge of town car park, and presumably plug itself in to charge while it waits.
The biggest advantage to self-driving cars is that you don't have to buy a car, so the car doesn't wander aimlessly, it just picks somebody else up and taxis them around.
A car, as currently used, is not just a transport mechanism. It's also a storage mechanism. (Umbrella, sunglasses, kids' car seats, emergency diapers, emergency cash, emergency feminine products, OTC meds (ibuprofen, antacids, etc), a reasonably secure place to keep my backpack if I don't want to carry it, mints/candy, etc.)
Automating a taxi service doesn't 100% replace the utility of a personal car.
Because of rush hour, a large fraction of self-driving cars will still need to park. However, they can drive themselves out of the city center and park in the cheaper outskirts.
Why would I want to use the same car as other people? It will be dirty, because people will do all sorts of things in there,and I would feel very uncomfortable.
In most cases that's not really an advantage at all. We already have a system like this: public transport.
Public transport seems to be generally rather bad in the US, which might make self-driving cars seem useful but other parts of the world often have better public transport and self-driving cars seem a lot less exciting here.
Annnd reality surpassed your idea already; some cars have a system using cameras and such that automatically parallel-parks it. One of my colleagues has a car like that, it's quite fun to sit in a self-parking car.
