The Massapequa Preserve is owned by Nassau County. Not sure who you would contact, why don't you try asking some of the cache owners who have hides there?
Jones Beach is a state park. You need to fill out an application for a permit and mail it to the Long Island Region Headquarters. They usually take about 2 weeks to get back to you.
Permits
Belmont Lake State Park
625 Belmont Avenue
Babylon, NY 11702
Edit to add: Tackapausha Preserve is operated by the Nassau County Department of Recreation and Parks. I believe some of the museum employees are Geocachers, you may want to stop there and ask.