As far as I can tell from the legislation, it basically gives the National Parks the power to create / vary traffic regulation orders pertaining to
"a byway open to all traffic, a restricted byway, a bridleway or a footpath, or a carriageway whose surface, or most of whose surface, does not consist of concrete, tarmacadam, coated roadstone or other prescribed material"
I think that previously the National Parks would have had no power to create / vary traffic regulation orders, this would have been responsibility of the Higway Authority for the particular road.
A lot of byways / bridleways probably don't have proper traffic regulations orders attached to them because of their historic nature, so who is and who isn't allowed on them is assumed in some cases. This means that the National Park will be able to create specific tailored orders for the above, effectively bannign / allowing certain types of traffic.
Theoretically, this could mean that they would have the powers to vary orders in order to allow motorbikes on certain sections of byways if they so wished (obviously unlikely)