It sounds like you will need to find a way to persuade your parents to support you going to NYC for college. Try reading online about persuasion techniques. You will want to avoid approaching it from a “I want to this” perspective and instead appeal to your parents’ wants and desires. Look at it from their perspective and think about their reasons for wanting you to stay. Then try to outline the main reasons why you to attending college in NYC is something THEY should want.
For example, if your parents are big on “maturity” and “responsibility,” you could explain to them how going away to NYC would be the best place for you to gain these qualities they want to see in their child. Or if your parents mainly want you to stay in Texas so they can talk to you all the time, you could appeal to that interest by reminding them how you’ll have so many unique experiences that you’ll want to call and FaceTime them all the time to tell them about your adventures.
Also, if you have to cave in and stay in Texas, there’s nothing wrong with that! Lots of people are moving from places like NYC and California to Texas.
Good luck!