I'd say your best bet for this situation is for your parent to file a formal petition for a name change in the New York Court system. The petition for a name change requires a the original or a certified copy of your birth certificate (available from the state of New York or City where you were born, google for instructions).
The reason for the petition would be that you are the person (mis)named in the birth certificate but you have always been known as [insert your name here]. Instructions and all the forms you need are on the New York State Courts website here http://www.nycourts.gov/forms/namechange.shtml .
While like any legal matter, it's possible for you to fill out these forms correctly and get the name change order you want from the instructions on the website, it would be a lot easier and you'd get the job done with fewer problems if you could retain an attorney to assist you in completing the forms correctly and filing them with the Court. If they are not done correctly, you will be retracing your steps and have to do one or more "redos". The Court Clerks can be helpful, but they can not give legal advice or answer detailed questions about how to prepare the forms.
An attorney should be able to accomplish this relatively simple job for a few hundred dollars, or, if you and your family cannot afford that, you may be able to get some assistance from a local Legal Aid Society. But it's not a difficult job, you just need to be persistent and correctly file the petitions and order for the name change by carefully following the instructions on the website and getting legal help if need be. Note that as a minor, you will have to have a parent or legal guardian participate in filing the petition. Best of luck to you.
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