Picnic in the Park
Q: Do I need a permit to hold a party in the park if there are more than twenty people?
A: Yes, any event with more than twenty people requires a permit from the Parks Department. You can also request a permit to reserve space in a park, even if you have less than 20 people.

Alcohol (unless special permission is granted) and barbecues are both prohibited in Central Park.

More info and a permit application can be found at the Parks Dept. Site.