I know you asked about national parks, but I thought I would add my experiences which are at state and city/county parks as well.
Most of the time the rule against sleeping in your car is to protect wildlife. If your sleeping in your car in a designated campsite for "RVs" then there should not be a greater impact by your van then a "streamliner". However, if your sleeping in your car at a parking lot, or, especially in primitive camp sits, the park services could get very upset.
As a general rule of thumb, you can think of your van as an RV and sleep in it just like you normally would an RV. If you are in an area where RVs are not allowed then you can be asked to leave, or in some cases face hefty fines.
One other thing to consider is usage. They will get very angry if you are sleeping in your van to avoid the in/out rules, capacity rules, or wildlife protection rules. For example, sleeping in your Van so that you can stay 16 days (14 at the campsite and 2 at the parking lot) will not go over well. Neither will sleeping in your van at a primitive campsite. They will likely have a problem with you sleeping in your van if you have the engine running all the time or are not properly disposing of waste. But if your at a campsite you reserved, in the time frame you observed, not trying to skirt around rules, and just using the van like an RV you should be fine. There are always exceptions though, so you best bet may be to call ahead and check with the ranger station.