Maybe they will. If not you can increase the tax even more and, worse case scenario they still don't care and you still get the money to invest in public transportation :)
Under capitalism, any resource that has to be limited due to scarcity or the need to cut usage (in this case, parking space/car usage in cities) will be used by the rich. It's just how the system works.
Charge for parking? The rich can pay.
Only allow cars with odd/even plates to enter depending on day of the week? The rich can have two cars, one even and one odd.
Only allow electric cars? The rich can more easily buy one.
Only allow to park at certain hours in the day? The rich can have more flexible timetables.
Only allow to park for a limited time? The rich can have a chauffeur to drive their car away while they do their business.
Don't allow to park at all? The rich don't even need to park, the chauffeur will drive their car away and that's it.
Don't allow cars to enter at all? Rickshaws will get popular.
If you don't want to drop or radically reform capitalism, and you want to solve this kind of problems at all, it's just something you need to accept. It comes with the system that with money you have more means to adapt to any restriction.