This is how it presently works in London, any roads on which pavement parking is permitted are signed and marked appropriately.I don't know what the answer is particularly but I do worry we're going to throw the baby out with the bathwater if we have a knee jerk reaction (i.e. I'm not sure anyone is served particularly well if we can no longer collect bins for instance!). Perhaps one solution to balance both users needs would be to mark on pavements (where the roads aren't wide enough to allow for everyone to park on the road without access restrictions) the limit of where a car can park.
In my borough, people can submit a petition to the council to request that officers review the situation. My council will only consider allowing pavement parking if there is at least 1.5 metres space remaining after pavement parking is taken into account. I'm not sure if that is a policy across the board.