Well, you don't actually ever have to park them. They could run near constantly, or any schedule that makes the most return on investment. Self-driving cars, if taken to their logical extreme, result in total gridlock.

I do think that between Ford and Uber, Ford is much better equipped to handle maintenance of a giant fleet of cars. Uber has zero experience in this area, whereas Ford has a giant, distributed network of dealers. If cars are predominantly owned by the company rather than being sold to consumers, these dealers might be in a position to pivot towards fleet management rather than sales and service.

However, I don't think any of this will happen for the simple fact that self-driving cars are not as close as people currently think. Maybe trucks and buses, but even then I remain skeptical.