Yes you need a park permit. Call Paramus city hall and ask about a permit. Its a County park so you may be referred. The Zoo is not always free so if you want to be away from that extra cost look at a map before you pick a picnic area.
Yes, for parties of 10 or more you will need a permit in the picnic areas, which can take up to 3 weeks to obtain. Check the Bergen County Parks website, it should be listed under FAQs or do a search.
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