Park hours are from 8am to sundown. You can fish from the banks with the required documentation from the Florida Fish and Wildlife Commission. More information at the link below https://myfwc.com/license/recreational/freshwater-fishing We are also known to the locals for catching snook in the Peace River which requires an additional saltwater license and a snook permit.
Sunrise to sunset give or take an hour or so and you can fish with a license
I believe you can. We saw fossil hunters but no fishermen. The park ranger was really helpful. Not sure about the hours.
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