In a recent ongoing discussion on RoW; half way through the thread 'nbt' commented
It is NOT a criminal offence unless a specific local bye-law has been passed, e.g. such a law has been passed within the peak district
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I am aware that the National Parks & Access to the Countryside Act, 1949 (as amended) and the Dartmoor Commons Act, 1985 make it a criminal offence to mtb where you shouldn't, within the Dartmoor National Park. This and other offences to do with horse riding, camping etc; due to the sensitive nature of the area (I assume).
I am NOT however aware as to similar legislation having been passed for any other National Park within England and Wales. Particularly the Peak District National Park which interests me the most. This is 'byelaws specifically aimed at cyclists' not motorised transport.