If you build new housing without parking for new inhabitants, you don't have that inherent excess of cars per parking spot. You might have some excess still but not nearly as much.
The biggest solution though is to offer alternative transport, first, and then removing, reducing or making more expensive the parking infrastructure. Here in Amsterdam the city has been removing parking and hiking up parking fees quite rapidly. But it was after establishing ubiquitous, separated, safe cycle paths everywhere, a decent tram, bus system and metro, 8 train stations (in a city of just 1 million) and another relatively high-speed underground light rail project in the works today, as well as public-transport's rentable bicycles and mini electric cars (which, due to sharing, is getting iirc about ~20 users for the space of one traditional parking spot). In Amsterdam I've never owned a personal car and won't ever need to. But having lived in America, I can't imagine that to work.
I thought we were talking about major and not-so-major cities; areas where there are mandates to install X units of parking per Y units of new housing. In places like these, unrestricted parking is almost non-existent.
When a university near me raised parking prices, any road or grass verge or dropping-off point on campus that wasn't marked 'no parking' got cars on it. When those were marked 'no parking', they started parking in the customer parking of nearby shops. When the shops restricted parking to two hours, people started parking on residential streets further out. So the residential streets needed a residents permit scheme.
Now, if parking is already extremely heavily regulated you might already have the street parking and customer parking and residents parking restrictions (and admin and enforcement for them) in place. But it's my experience that restricting the supply of parking spaces isn't always a simple matter.