Upvote:8
If you are prepared to pay β¬15/day, you might just as well stay at a proper campsite (with toilets, water, etc.) I don't think booking car parks in advance is generally possible (never noticed it in the resorts I went to but I usually stay away from the biggest, most popular resorts) but it might be a good idea for campsites.
Some potentially useful websites:
I have never done it myself but judging by these websites, it seems there are many options. I don't think the distance from London is going to help you choose either, most ski resorts in the Northern part of the French Alps should be reachable in 12 hours, the difference between the furthest and the nearest is probably not more than 1 or 2 hours.
Note that in France, municipalities can forbid camping or staying overnight on car parks if they see fit and I don't think than sleeping in a van instead of a RV makes a lot of difference.