I don't know about the NH state parks, but the National Park Service spells out pretty clearly
when permits are needed, under the general category "Commercial Filming and Still Photography Permits". Basically if it's a location not normally accessible to the public; you bring in models, sets or props; or the park service would need additional resources to monitor the activity. He's bringing in a costume, and he's doing it to advertise his other artwork, so it would probably require a permit in a national park. But small scale, there are no onerous fees: 1 - 2 people, camera & tripod only $0/day. The system is set up to keep advertisers and corporations from abusing the parks for their own uses.
In the article, it sounds pretty similar for the NH state parks, except the fee is $100/day. As a photographer (who spends time in CA state parks and national parks), it doesn't sound to me like a question of free speech because they didn't deny him access, they just told him to follow the existing rules and get a necessary permit.