Answer:
Yes, but only in non-residential locations that are in designated areas in city parks:
NYC Parks & Recreation BBQ list by borough:
http://www.nycgovparks.org/facilities/barbecue
Roosevelt Island:
My favorite place to BBQ is on the northern-most tip of Roosevelt Island in Lighthouse Park (by the historic lighthouse). The area overlooks all of Manhattan and is a beautiful place to BBQ at night.
http://rooseveltislander.blogspot.com/2011/08/roosevelt-island-lighthouse-park-under.html