I've called Alum and Buck, and nothing is needed for Alum creek at the beach as the Monk states. Buck wants you to have a permit only if you want to search non-beach areas.
Long of the short, the state parks just don't want you to prevent others from enjoying their stay at the park.
Just call the park in question and ask to speak to the Ranger on duty. Tell them about your hobby and that you would like to know what you should do in order to obtain your permit (they're free). Also ask them to lay out the specifics of where you cannot detect. In the case of the picnic areas, ask them specifically if it means that you cannot detect there at all or just when people are picnicking? My answer was that I could detect as long as no one is there, but that I needed to make sure that I had my permit and that I stopped by the office (if it was open) to tell them that I was there (John Bryan park). The other parks didn't care because they were beach parks. No permits or documentation to take with you.
Again, it is always in your best interest to find out what each park manager would like you to be able to do on their property. The beaches to the bouey, and not when it is busy (ask what "busy" means).
PS I'll be putting my Tiger Shark to the test here real soon at Buck Creek, and testing their "don't worry about stopping by the office" comment.
After waving my detector over my wife, I can honestly say that I detected a Heart of Gold!