Instead of crescent you may want to try Kettle Cove. People have their dogs on the grassy area by the beach all the time. If you follow the path all the way past the point there is a small beach there that is secluded and usually empty. We let our dog swim there in the summer and have never had issues.
Dogs are not allowed on State Park beaches from May till October. They can walk on the trails on leash.
Dogs are not permitted on Crescent Beach but they are allowed in the picnic area and there is a pet trail on the right just after entering park. If you want a dog-friendly beach, I would suggest Pine Point Beach after 5pm or Higgins Beach
I think done are totally banned until September 30, but are allowed at the the rest of the state park.
Dogs are not allowed during the day. Most beaches allow dogs before 9 am and after 6 pm (like Pine Point).
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