A few observations and bits of input from someone who drives for a living:
I always avoid parking on pavements where possible, out of respect for pedestrians and wheelchair users etc. Even if 2 wheels on the pavement will allow traffic both ways to pass simultaneously all 4 wheels stay on the road and they can wait / give way as appropriate.
If 4 wheels on the road means a total road blockage but 2 wheels on the pavement will allow 1 vehicle at a time to pass then 2 wheels will go on the pavement whilst leaving the maximum possible width for pavement users and making it a tight squeeze for road traffic.
Lots of new build estates are a nightmare with totally inadequate parking for the number of vehicles. Residents block pavements completely, badly parallel park at angles leaving bumpers sticking out, park right at corners and junctions meaning some shunting back and forth in a large vehicle is needed to get round the corner or turn at the junction.
It's a similar situation on straight streets with, say, Victorian terraced properties. Indeed in Oxford (and I dare day other cities too) some such streets have parking bays marked partially on the pavement both sides of the road to leave room for single file traffic down the middle.
As others have eluded to, unless the number of vehicles on our roads reduces significantly (which I don't see happening) then pavement parking is here to stay. I don't think the views of those against pavement parking would be anywhere near as strong if drivers, when it is necessary to pavement park, were more considerate in how they do so.