It's something that could go awry quite easily, if implemented poorly. However, some of these things are actually much costlier than people realize and they might not exist if people actually had to pay for them directly.

City parking, particularly streetside, has an almost unfathomable opportunity cost. That real estate is very scarce and there are much more productive things that could be done with it than just having someone's car sitting there.

That said, there are all kinds of other regulations (zoning and the like) that prevent alternative uses and create monopoly situations.

The most straight forward answer to why it would be better to privatize roads is the same as for everything else: the state is a monopoly that produces things far more expensively and at lower quality than could be provided on an open market.