(In my (European) city we actually have the opposite - new developments are unlikely to be approved unless they're "zero parking", because we already have as many cars as the streets can take).
Taxpayer owned, profits (after maintenance, expansion of the program, etc) going to decrease taxes, parking structures at the edges of cities located at nexus points of vehicle and intra-city transit grids: preferring density on both.
Step 2:
Make movement /intra/ city on public transit and systems free (possibly only for uses of the parking system).
Step 3:
Affordable 'carts' designed to be used with the transit system for people to move goods from their cars and/or shopping trips back to their homes. Bonus points if there are 'pod' versions that are lockable, trackable (smart phone app), and which other delivery services can be paid to bring to your apartment/condo/etc.
Step 4:
Expand the jurisdiction limits of the city to the point that it encompasses the metro area, to make planning actually work.
Step 5:
Encourage development of the right kind of housing in the right areas; encourage it by carrot and stick until the price of housing is where you desire it.
Of course, the parking lots service ~1000 people, while apartments obeying the height limits might accomodate a few dozen and would of course be even more desirable/expensive now that commuting to the station from far away is harder.