the city decides things like street parking, minimum parking space requirements, or if you can put up a parking lot, thus deciding the total amount of available space. actual operation of the parking spaces can be offloaded to private entities.

if less space is allocated, it becomes more expensive (scarcity), driving down car use. this is an easy case of pricing in environment and external impacts.

Yes, that can happen. Then another city council gets in, and wants more parking.

The post I replied to said "The whole point of charging for parking" was to force people to, essentially, drive less. This is saying it's the only reason people charge for parking. Clearly not so.

And there is artificial scarcity too. I've seen cities where one company owned almost all the parking lots. In that situation, they controlled pricing entirely.

I've seen a lot more of "doing it for profit", including the city doing it for profit, than for environmental reasons.

That's doesn't mean I support this position or not. Describing reality as it stands is important, one cannot effect change one way or the other, without understanding what is.