One of the problems with allowing parking on the pavement is by implication you allow driving on the pavement. And that can be very dangerous.
Take a look at this picture of Shaftsbury Road, Sutton:
It's a relatively quiet residential road, and on such a road you would expect pedestrians, including young children, to be able to walk with safety on the pavement. Not so here. If a driver is in the silver car nearest the camera, the elderly gentleman may be in his 'blind spot'. If he chooses to reverse into the road, he may hit the pedestrian. Remember, the driver's attention may be focused much more on dangers on the road than on the pavement. If small children emerge from the gateway of a house, they may not be visible through the rear window of a vehicle - particularly if that vehicle is an SUV (Jeep/Land Rover) or a van. Like for instance the two vans you can see parked on the left-hand pavement. When you teach young children road safety, you teach them to stop at the kerb and not go into the road. The kerb is recognizable even to very young children, which is why you don't usually see kids running into the road where they would be in danger. Take away that barrier and you make life very difficult for parents. Will a responsible parent wish to let their children run (or scooter or cycle) down this street, with the constant danger of cars manoeuvring on and off the pavement?
In Sutton, the rights of children take second place to the convenience of motorists.