Most national parks allow dogs to come into the national park with you even if they are not service dogs as long as the dog is a friendly dog and is on a leash at all times.
If you require a service dog and the national park says no dogs allowed in the national park then yes you can bring your service dog into the national park with you.
The service dog must be a registered service dog and be doing a service for you and the service dog must be allowed into the national park or other public places that you go to.
You cannot be discriminated against when you have a disability and that goes for having to have a service dog as well and no place even a national park can deny you entry into the national park when you have a service dog.
The service dog must be allowed into the national park when you need the service dog.