Motor-driven cycles issued a plate or device pursuant to Section 38160 may be moved, by nonmechanical means only, adjacent to a roadway, in such a manner so as to not interfere with traffic upon the highway, only for the purpose of gaining access to, or returning from, areas designed for the operation of off-highway vehicles, when no other route is available. The Department of Transportation or local authority may designate access routes leading to off-highway parks as suitable for the operation of off-highway vehicles, if such access routes are available to the general public only for pedestrian and off-highway motor vehicle travel.