https://lwd.dol.state.nj.us/labor/lsse/laws/Model_Rocketry_Act.html#1c2 is the law governing model rocketry in NJ.
This is what the law says about permits.
SUBCHAPTER 5. PERMITS
12:194-5.1 Permit restrictions
A person shall be required to obtain a permit for the storage of more than 100 kilograms (220 pounds) of solid propellant model rockets. No other permit shall be required for the possession, use, purchase, transportation, or sale of model rockets.
However NI has adopted NFPA 1122 as shown below.
12:194-1.3 Adoption by reference
(a) The standards of the National Fire Protection Association (NFPA) 1122, "Code for Model Rocketry," 1994 edition and subsequent amendments thereto are hereby incorporated by reference as the rules for the manufacture and use of explosives in model and amateur rocketry. Copies of the NFPA 1122 may be obtained from the U.S. Government Publications, Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402.
(b) In addition to the standards incorporated by reference, this chapter sets forth additional rules governing the use of model and amateur rocketry.
NFPA and the NAR Model Rocket Safety Code also requires that
you have the permission of the land owner to launch model rockets. While it is a public park, and although it may not be against any written park regulation,
it is always wise to get a permission letter from the head of the park so that if a ranger comes up to you, you can show him the letter.
If not explicitly forbidden it's usually not hard to get a permission letter if you know the state and local laws, review the safety information on the NAR website, and be able to answer questions about the safety of model rocketry. You stand an even better chance to get permission if you are a current NAR member, you can show him your model rocketry "license", the NAR Model Rocket Safety Code, a copy of "Launch Safe" and a copy of the NAR insurance policy which covers you up to $1,000,000 if an accident should occur provided you followed all Model Rocket Safety Code Rules. (Municipalities really like insurance policies.)
If you look and act responsibly, and can answer questions about the hobby and it's impeccable safety record, you should be able to get a permission letter.
References:
https://www.nar.org/pdf/hobby_overview.pdf Historical safety record of model rocketry
https://www.nar.org/cabinet/MRSafetyCode.pdf Condensed version of NFPA 1122, the state of NJ regulations
https://www.nar.org/pdf/NAR%20Member%20Insurance%20Certificate%202009.pdf Copy of the individual NAR member insurance coverage.
https://www.nar.org/pdf/launchsafe.pdf Something everyone should read.
https://www.nar.org/pdf/Safety_in_Sport_Rocketry_Tutorial.pdf Something else everyone should read.
Bob