Only legal parents have rights. If you have no legally recognizable proof that you are the parent of a child, you are considered a "legal stranger" to the child and have no parental rights.
This folder contains materials that explain how to establish Paternity - how to become a legal parent.
Paternity Proceedings in New York State: The Basics (Cross-Borough Collaboration)
Follow-Up Legal Action Plan - Paternity (Reentry Net)
NY Family Court Forms - All Paternity Forms (NYS Office of Court Administration)