Picnics of more than 20 people require a permit through the NYC Parks Department. You can obtain a permit here.
Picnic tables are located by The Great Hill (West Side from 103rd to 107th Streets) and the Ross Pinetum Playground (West Side at 85th Street). These locations are in high demand and cannot be reserved, so they may not be available.
Tables and chairs are not permitted on grassy areas due to the damage that the legs may cause.
Alcohol is not permitted in Central Park, except at specific restaurants and their patios. See more info about alcohol here.
Barbecuing, open flames and grilling is not permitted in Central Park, with the exception of 3 days per year - Memorial Day, Independence Day and Labor Day (just the day, not the weekend). The Harlem Meer or the Great Hill will have smaller crowds. Propane use is not allowed. Find out more information and rules on how to BBQ in Central Park here.