Here in Amsterdam we also have an increase in car-free zones, but one effect of that is just to increase car parking pressure in the neighborhoods of those of us who live adjacent to the center.
I think the real solution is to end socialism for cars. This problem will solve itself as soon as the land being used by parking spaces isn't specifically zoned, and thus would need to compete price-wise with using the same amount of land for residential developments or shops.
It's also interesting to see how this breaks down as a political issue. E.g. in Amsterdam the generally free market party (VVD) has a notable break from their "their market will take care of it" policy of wanting more government-subsidized car parking, v.s. the leftist parties.