If you need to go to an area closed to the public, or be there at a time when an area is closed, of course you need a permit, but so would you even if you were not photographing. Generally speaking with editorial and fine-art work, there is no need for a permit, because you'd be photographing only what is found in the park, and not using a crew. Even if your photos are going to be used as commercial stock, as long as they are taken in accordance with the quoted paragraph you don't need to apply for a special permit. You also do not need any type of property release, as long as you're not photographing a private property inside the park. Releases protect a person's right of privacy.
Public lands have no such right.
On the other hand, some types of commercial photography requires a permit. This is loosely defined by the NPS as anything which requires you to have a model, crew or props, and includes wedding photography. The rules were mostly intended towards motion picture production and commercial product photography, which can cause a significant disruption to sites. If you want to run a workshop in a NP, you need a Incidental Business Permit, since this is considered commercial usage. Such a permit is quite strict regarding your number of participants as well as itinerary.