Did your father lose Custody or have his rights terminated? There is a big difference. In any event, he'd need to Petition for a change in Custody (assuming his parental rights were not terminated), in the jurisdiction where you live. He's not going to get anywhere filing in TN unless the Custody order was issued there. Even if it was, he'd likely end up litigating it in NY if you've lived here more than six months.
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First lets tackle where? If you have lived in NY more than 6 months then NY has jurisdiction so he would file in NY. Now the what. You say he has "no pernatal rights". I assume you mean parental. If you mean his rights were completely terminated then he may have a problem prior to your turning 18. If you mean he lost custody and your mother has full custody, then he can petition for a change in custody. Be aware, however, that just because you want to live with him does not mean the court will grant him custody. It is what the court believes is in your best interests. However, based on your age it is a very good possibility. But first you must determine what you mean by he lost parental rights.
The above is a general answer and is not considered legal advice. You should contact an attorney before proceeding to take any legal action, signing any papers or upon service of a summons. Howard E. Knispel 631-864-7589
I never want to suggest self-help, even to a 16 year old. It can have negative legal consequences against your father. However, if he comes to NY and files a petition, he can get a temporary order of custody and you can move quickly---if your mother does not oppose it. Have you discussed with your mother this desire to move? Maybe she will agree and everything can be done in one court appearance.