In my experience.... Not really. On the bike getting into the park is cheap and not that many parks have entrance fees. In washington I can only think of two.. Olympic National Park and that is only to go up to Hurricane Ridge(all other entrances into the park are free) and then Rainier National Park. Both have a fee of I think $5 or maybe it's $7? for a bicycle and that's for a whole week last I checked.

YMMV.