This is confusing National Trust with PDNP. They are different things, although they both own or control land in the Peak District.
The PDNP have byelaws that very clearly ban model aircraft and drones from “open access land”.
See byelaw 17(ii).
You can find maps of the Access Land in this area here:
http://www.openaccess.naturalengland.org.uk/wps/portal/oasys/maps/MapSearch
If it is Access Land, and within the PDNP’s overall boundaries, it’s covered by the byelaws.
Having said all that, I’ve flown within this land and nobody seemed very bothered!