Is it allowed to launch powered paragliders from national parks?
FAA has the following informational web page for Alaska National Parks:
https://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/systemops/fs/alaskan/advisories/parks/

Which has contact information for parks where flight operations are periodically done.  Several of my pilot friends have used park and forest service airstrips with permission, so it is worth asking.  In Alaska, the parks are generally more open to aircraft, than in the lower 48.

There is also an advisory circular, 91-36C.  https://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/systemops/fs/alaskan/advisories/parks/media/ac91-36c.pdf

Some of the lower 48 have strips where permission may be granted, plus forest service has some strips (usually gravel) which permission may be granted to use.  Again, contact the park in question, because they often have local rules which are permissive.

State parks vary. Some states prohibit aircraft of any kind, others permit operations, usually situationally.  

Paragliders are just one class of airman who are impacted by park policies. sUAS pilots are also impacted.

Regardless of what you are told verbally, I would only consider ops in a Congress Designated Wilderness with written permission. Those areas have a prohibition of mechanized devices, and that includes hang glider and sUAS activities.