> One could prevent less parking to be built but that would put more pressure on public roads.
Yes, and if demand on public parking exceeds supply, you just do the obvious: charge for it until you hit equilibrium.
> If you've ever been to an area with insufficient parking, it's pretty horrible.
It's actually not, you're just thinking with a very American mindset, assuming that everyone will drive everywhere, all the time. I recently moved to Munich, a place that, by American standards, has relatively little parking. It's fantastic, because you can actually walk and bike and take transit places, and it's comfortable and convenient.
Now I know what you're thinking: what does that have to do with offering parking? But the thing is, land use is mostly zero sum. Forcing everywhere to have huge streets and parking lots means that everything is spaced further out; not a big deal when you're in a car, but a very big deal for walking, biking, and transit. Requiring parking helps cars at the expense of other modes.