Their best bet is to inquire at some of the parks they are considering to see if commercial vehicles are allowed, particularly if they want to overnight
Yep. Federal parks roads are administered differently than the public roads the FMCSA deals with. In most cases the roads are Army Corps of Engineers roads. Even the very definition of a Commercial Vehicle is different within federal lands versus the public roads. If you're passing through a national park in a station wagon, and you are engaged in business, you're a Commercial Vehicle.
Yellowstone, for example, I know limits commercial vehicles to trucks and vans up to one-ton in size, 4 axles max. Tractors and straights are not allowed, even with the required Day or Annual Commercial Permit. Some parks will allow up to two or three axle trucks which are able to transport up to 54,000 pound loads, with a permit (Rocky Mountain National Park in Colorado is one). Others won't allow even U-hauls because of the advertising. There might be a national par that allows a tractor trailer to use the roads, but I'm not aware of one.