The problem is that on some streets pavement parking makes perfect sense. There are some around here where the pavements are wide but the roads narrow so cars park with two wheels on the pavement, leaving plenty of room for a wheelchair.
On my parents 1930s road, there isn't enough off road parking for the standard two cars per household, so they park on the pavement. There's room for a single pedestrian but no more - but it's a quiet residential cul de sac so everyone's happy to walk in the road if they need more space as there isn't actually that much moving traffic.
Essentially, the law needs to say "don't be a dick for this particular road at this particular time" but that's hard to legislate for.
(People who block the pavements on busy roads should be in the stocks.)