You are asking about dispersed camping which is legal in state and national parks. However I would still call the park office before doing so.
No but they have cabins and places for tent camping and campers.
There are no campsites along the Platte River at River State Park
Yes and the only cost is a state park permit. Keep in mind the river rises and falls 1' everyday because of a power plant upstreams water usage and any rain upstream may unexpectedly flood a tent. Dispersed Camping is AWESOME! Say no to glamping!
I've done it. Just make sure it's not on one that gets submerged at night
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