If they are going to accept that pavement parking is the lesser evil in any particular solution, it would be better, in my opinion, for them to narrow the pavement. Then you can provide a consistent message that pavement parking is unacceptable everywhere. It is also significantly friendlier to pedestrians, because it completely maintains the level separation between vehicles and pedestrians.
I would be in favour of the installation of devices on the kerb to make it difficult to drive onto the pavement, but that might negatively impact wheelchair users so would not be ideal.
In many situations it may even be possible to entirely remove the pavement from one side of the road. Why is it necessary to have a pavement on both sides? (Says the person who grew up on a road with a single-side pavement, which no-one ever used. Admittedly the village green was on the other side of the road!
) Personally I'd have absolutely no issue with crossing the road to get to the pavement on the other side. Although obviously it does depend on local circumstances: how busy is the road? is there a large volume of pedestrian traffic? etc etc.
Take for example this road:
https://www.google.com/maps/@52.099...4!1spqv9Kme8Z_VwJh0drKe_Dw!2e0!7i16384!8i8192
A row of Victorian terraces, with no driveways. The only place to park is on the road. Unfortunately, there just happens to be a large school at the end of the road (behind the camera, if the link works correctly). Pavement parking is therefore a definite no, because of the large volume of children arriving every day on foot. However, there also needs to be a gap large enough for around 13 buses/coaches to travel both ways, twice a day, within around 20 minutes, as well as all the car traffic in the area. The road is wide enough for a single lane of traffic, but all hell breaks loose when some idiot in a car goes head-to-head with a bus... If you're fortunate, there will be a gap in the parked cars. If not... well, let's say I've spent a good few hours on coaches not going anywhere on that road!
(Incidentally, I could create a whole new rant thread about traffic in Evesham!)