That's assuming there is a monopoly on providing parking spaces. Zoning notwithstanding, anyone can build a parking structure and rent or sell parking spaces. Tenants can choose whether to pay more for an apartment that includes parking or less for one that doesn't.
The total cost of a housing unit with parking wouldn't be be more than it is now because the cost of constructing a housing unit with parking wouldn't be more than it is now. It would just cause units without parking to become available.
Which would tend to increase housing availability, which would tend to lower prices including for units with parking. Because the people who want the parking spaces wouldn't have to bid against people who just want housing and don't need parking.
Fun fact - the city near me has a new zoning rule that limits the total number of parking spots in the city to something close to the current number of parking spots. If you want to build an apartment with parking, type find to need to tear down something with parking first.
So they're tearing down shopping centers, movie theaters, and grocery stores just so they can build apartments with parking. I have no idea where the people living nearby that don't have cars will buy food, but at least the new luxury apartments will have parking.
The total cost of a housing unit with parking wouldn't be be more than it is now because the cost of constructing a housing unit with parking wouldn't be more than it is now. It would just cause units without parking to become available.
Which would tend to increase housing availability, which would tend to lower prices including for units with parking. Because the people who want the parking spaces wouldn't have to bid against people who just want housing and don't need parking.