If the neighbourhood isn't commuteable without a car, the worst-case scenario is to the remove parking and expect everyone to just manage.
If parking is truly an issue, the market can of course solve it with private parking lots.
No, it can't - it doesn't.
I think some supporting evidence would be good in your post. Is there some kind of market failure at work here? Absent regulations preventing it, I see no reason people couldn't build a parking garage or something. Here's a parking garage in the oldest, densest part of Padova, where I used to live: https://goo.gl/maps/sS3dds2h3yt - it's not cheap (they use an elevator system!) but someone built it and operates it at a profit.
"Someone built an expensive garage" doesn't demonstrate solving anything.
It is "not being solved" in a system that bears little resemblance to a free market. Housing and cities are one of the most tightly regulated markets in the US. The level of detail that most zoning codes go into in the US is mind boggling.
For instance, with regard to parking minimums, here in Bend, we have this (which was probably cut and pasted from some other town):
> "Beauty parlor and Barber shop: 3 spaces for each of the first 2 beauty or barber chairs, and 1½ spaces for each additional chair"
That is most emphatically not what I would call a 'free market', but rather detailed and specific government regulation.
I'm not a libertarian and I do not think markets solve all the world's problems - far from it. However it is undeniable when you start looking into the details, that these markets are very, very far from "free markets".
Converting parking lots to apartments just gives you more apartments on the same footprint. It doesn't bring the workplaces of the people in those apartments any closer.