The issue is not parking. The issue is that these cars remain on the road all day, creating congestion. Commuters are only clogging the roads during commutes. Thus “rush hour.”
The question was why they stay on the road when they're not delivering someone. If the issue isn't lack of parking, what is it?
The commuter and residential parking doesn't free up for Uber drivers if it's not being used by its primary customers.
Disclosure: Extrapolating from how it works in my town.
If they travel by taxi there's only one car on the road - but the taxi has to do a third trip, going back from the city to the suburbs between passengers. That trip is likely to be made without a passenger - because in the morning rush hour, far more people want to go in one direction than the other.
Of course, in some cases taxis may help reduce congestion - transporting people to and from train stations, and allowing people who usually cycle to transport bulky items, might enable more car-free living.
Cities already have absolutely massive amounts of their real estate dedicated to parking. The answer to this problem is not and never will be “more of the stuff that’s causing the problem”.