I think, generally speaking, discretion is used. A tidy campervan or motorhome parked up in a layby for one night may be tolerated, where a group of Transit vans pulling twin axle caravans would not be. Doesn't make it lawful though...
I've just has a look at the petitioner's website, the petition is aimed squarely at getting the same rights for England and Wales as now exist for some access land in Scotland for people accessing the land on foot to camp with a tent or bivvy. IMO it's a very reasonable request and, because of other interests I have, I will sign it.
However, even if the petition does result in a change in the law, it is extremely unlikely (and is not the intention of the petitioner) that any resulting legislation will give any similar rights to motor vehicles.
AndyC