Parents don't want me to take out loans?

Or be prepared to have to drop out of school if even the slightest glitch happens. If only an $8000/ yr difference exists, the OP did not apply to schools that are actually affordable. But that is a totally different issue. If her parents are unskilled laborers, an accident, illness, a layoff, or reduction in hours, or necessary major repairs to their home could mean the difference between being a college student and not.

It means no room for a single financial crisis over a 4 yr period.

If the parents are laborers and are paid under the table, they may not be contributing to SS.

The OP got almost a full tuition scholarship from the local private university that is one of the best in the field. How much better could she do? Maybe Columbia would be a bit cheaper. Any more affordable options would imply a full ride.
Can you commute to NYU from home if need be? If yes then Stern is clearly the best choice.