Both the mother and father have equal rights with respect to custody unless and until there is an agreement or court order delineating who has custody and who has parenting time when. If the father's name is not on the birth certificate he can still establish paternity by signing an acknowledgment of paternity or filing in court for paternity and taking a DNA test
Agree with Ms. Silberstein with one addition: the father should submit forms to the "putative father registry" which is maintained by the State of New York to secure his parental rights. I'm attaching their web brochure which discusses the forms he needs to submit and how the program works. Please give the father this brochure or link to the website.
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You and the child's father have the same custodial rights regardless of whether the two of you were married or if the father's name is on the birth certificate. Without a court order or agreement, you will not be able to dictate a parenting schedulle for the child. You should have an order or agreement in place that you can enforce and also receive child support.
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