Here is something to read before you apply to NYU: http://www.theatlantic.com/national/archive/2013/08/the-expensive-romance-of-nyu/278904/ It’s a couple years old, but I suspect that many of the points are still valid.
You probably need to have a discussion with your parents sooner rather than later about money and the cost of your college education; which should include a discussion of potential costs of graduate education (especially if your undergraduate degree is in English or a social science).