FWIW I live in an apartment building, about, 35 units, built c. 1970 with one parking space per unit. I regularly see my neighbors' second and in a few cases third cars parked on the streets nearby. So yeah, putting up apartment buildings without parking spaces will create parking problems, but those problems are probably there already in a lot of places where there are no buildings without parking spaces.
In urban areas cars seem to max out whatever infrastructure is created for them. And if all the bikes in the US were to disappear overnight, the same streets, highways, and parking lots that are congested or maxed out now would remain congested or maxed out, if not even worse.