There are no regulations that prohibit operating an ultralight (or any other aircraft) over a national park. We have a similar question about Yosemite and the same general answer would apply to any other national park.
For ultralights specifically, the regulations are in 14 CFR 103 which has only a few very simple restrictions on where ultralights can operate. 103.15 says you can't operate over a congested area or a crowd of people and other restrictions are the obvious ones: don't fly near airports without ATC approval, don't enter restricted airspace, and do follow NOTAMs.
That means that - as with the Yosemite question above - it comes down to whether the national park in question is a congested area or not. Realistically, most parks where you might want to fly an ultralight probably aren't, so it's very likely that you can fly over them at any altitude you like: part 91 doesn't apply to ultralights, so the minimum altitudes mentioned in the Yosemite question wouldn't matter.
There's also AC 91-36 to consider, which requests pilots to stay voluntarily above 2000' AGL. I don't know whether it's meant to include ultralights (it's a part 91 AC) but it doesn't really matter because the general guidance is definitely applicable to anything that flies:
The intent of the 2,000 feet AGL recommendation is to reduce potential
interference with wildlife and complaints of noise disturbances caused
by low flying aircraft over noise-sensitive areas
As you already mentioned, there might be restrictions on taking off and landing in a park or other public areas (that's an issue for seaplanes, helicopters and bush pilots too), but those would be non-FAA regulations.