Parking minimums prevent developers from free-loading on a commons, that commons being street parking.
So eliminating parking minimums by themselves will create nasty side effects.
But of course the correct answer to tragedy of the commons is pricing -- price the street parking appropriately and it won't be abused so you won't need worse solutions like parking minimums.
If a developer builds in a way such that the demand for street parking outstrips supply, the street parking still has a cost, that cost is just expressed in time to find a spot, not dollars like you're suggesting. People unwilling to pay that time cost will find paid lots or not have a car (which is basically the dynamic in my building: people either pay $450 a month for a spot or they spend 10-15 minutes looking for a free street spot).
In practice, of course, existing residents feel entitled to "their" street parking and get mad when a new building with new people contending for those spots is built but there's no logical reason to preference residents who have previously lived there. This is where politics rears its head though.
If we're talking about commercial properties and zones, people unwilling to pay that time cost just won't come to the area.