TL;DR, not in areas with other bathers. Most trails are open to your dog being on a leash. Dogs aren't really allowed on camp grounds or on the beach of the Atsion Rec Area, stuff like that. So no swimming with other bathers. Again you can hike with them about everywhere. If you're asking about the trails out there... I doubt park police care, but that needs to be taken with a grain of salt, since I don't actually know. Mostly the police don't want anyone getting hurt, or making a mess of the park. New Jersey is really weird about being "allowed" to swim in ponds or in rivers without a lifeguard on duty. For example, you can kayak down the Mullica freely without supervision, all the way to Atlantic City, etc., but park police will probably get on you for swimming somewhere without proper supervision or a PFD or something if they catch you... also snapper turtles and beaver are a thing out there. So be careful. Be safe. Leave no trace.
If your talking about Lake Oswego, then yes
Where exactly do you mean?
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