I can think of no reason why the council can't start charging for the use of the car park. There are laws which give people certain rights to take possession of land (through 'adverse possession') or to have access over it (through the designation of a public footpath) if they've been using it without challenge for many years but such laws simply aren't relevant here.
It would seem to be no different to what many councils do with street parking in residential areas, which is to designate the spaces as being for the use of resident permit holders only but then to charge each of those residents hundreds of pounds per year to have a permit. That's completely legal and I can't see why charging for the use of a council-owned car park should be any different.