I lived in NYC over 20 years ago and I go visit every couple of years, and I'm struggling to think of a regular restaurant that would accommodate such a large group. You almost need a private room. I do remember going to a large birthday party at Dos Caminos (an upscale Mexican restaurant). They used to have a few restaurants in the city but not sure where they are anymore. I'm not up to date on "new" restaurants.
(I loved Pio Pio but the one closer to where I lived was so tiny).
Rooftop bars will have an age restriction especially at night. I loved 230 Fifth (I think that's the name) but I would reach out to confirm age restriction or if there is designated time allowed - maybe for lunch?
For fun places to take photos (for teens), I'm thinking: Times Square, Rockefeller Plaza, Grand Central, Vessel at Hudson Yards, lions at the main NYC Library at Bryan Park, stairs of the Met Museum (which is next door to Central Park). While on the UES, cute street on UES (I'm thinking mid 60s St between Fifth Avenue and Lexington). Central Park is gigantic and you would need somewhere that is close enough to the street. Maybe Fifth Avenue at E75th, or on 59th Street.
I think a group of teens would love to visit SUMMIT observation deck. My daughter enjoyed it much more than Top of the Rock at Rockefeller. It wasn't cheap but it was fun (and lots of photo opportunities in there!).
Ps. I like the idea of a water taxi/ferry someone recommended.
EDITED: 230-fifth allows minors with parents but before 7pm. It was brand new when I was there in 2007 and they seem to have expanded quite a bit. Check them out. https://230-fifth.com/info/