Parks...........
While I can't give you any definitive info on allowing bikes in parks, I find it hard to believe they would not be allowed on paved roads.
The reason I posted was to say that 'Topo USA' is a great resource for looking at roads and elevations, for planning trips.
Also, as I looked at the NPS site I noticed that there is a way to contact them with questions, so you can try that.
Lastly, you can check in the 'Regional' forums to ask about specific parks. Good luck.