For what it's worth, the parking pre-dates the crossing by decades. I questioned with officers whether this was the best location for the crossing at the time it was installed.

As to the legality of crossing pavements to park, I'm not sure that the presence or not of a dropped kerb is the relevant consideration. However, I'm sure the lawyers will know. In general terms, I believe it is legal to cross the footway to reach a parking place on private land.