Many parks have a tenting area that is grass or other ground cover and an additional parking area for the vehicle. You would likely NOT be welcome to park your car or van on those tenting areas. Doing so would subject the area to ruts, to any noxious drippings from under your vehicle such as oil and grease etc. Nobody wants to pitch their tent on top of sludge from the underside of a vehicle.
You may be able to stay in your car in the parking spot, but that will vary by establishment. Many places just say no because to many people sleeping in your car reeks of homelessness, which is not an image businesses want to project onto their customers.