Not sure about that, in the UK the National Parks don't own any land to speak of, other than perhaps some car parks and visitor centre if there is one.
The National Trust do own some tracts of land in the National Parks (but most is just normal private owners e.g. farmers ). As mentioned they have no jurisdiction to prevent you over-flying but they can reasonably ban taking off from their locations.
As you say, the National Parks are more administrative areas and are they don't actually own the bulk of the land other than their own offices and some car parks. Most of the land within National Parks is in the hands of either private landowners (including the National Trust) or the state.
The NT does own a *lot* of land, including large areas of some National Parks, but in many specific cases within National Parks it can just be a car park and a small surrounding area. In those instances, provided you don't take off from the car park or other NT land you're good to go, one great example of this being the White Cliffs of Dover where a strip of land at the top of the cliffs is owned by the NT, but the beach below is not. Take off from the beach, avoid the NT land and the Port of Dover, remain in compliance with the Drone Code and you're fine!
If in doubt, the NT does at least maintain a very accurate map of the land they own or just manage (where you may still be OK to fly), including overlays with additional info about the type, ownership, when it was bequeathed/bought, etc. Click on a given area for more info.