No there is no way to give up your parental rights voluntarily, unless the child is adopted by someone who agrees to take on your financial responsibility.
If you want visitation rights, contact an attorney in the state where the mother and child live. Undoubtedly there is a way to do something and that is the state where that sort of litigation would need to occur. The reason your state had jurisdiction for support is that that is where you live, and you are the payor.
Please note that I do not practice in your state and that the above is not intended as legal advice, it is for educational purposes only. No attorney-client relationship is created or is intended to be created hereby. You should contact a local attorney to discuss and to obtain legal advice.
Respectfully, I somewhat disagree with the first answer. In WI, at least, you can do a voluntary termination of parental rights here. While courts in WI do prefer to have another parent ready to adopt in your place, there are some circumstances in which a voluntary termination may be granted without an adoptive parent.
One example was two parents fighting over custody, placement, and child support for over a decade. There were two children and two parents. The court agreed to allow each parent to take full responsibility for one child and volunarily terminate their parental rights to the other child.
So you see, it can be done without an adoptive parent, but the circumstances have to be pretty extreme.
Unfortunately, you do need an experienced family law attorney where the mother and child live. An experienced family law attorney should know whether a voluntary termination is available to you in Ohio.
I leave you with this thought: Is this really in your best interest/the best interest of your child? Perhaps, you should fight for your rights in Ohio rather than just giving up. You may even be able to have the mother held in contempt and punished for her actions.
Please understand this is a general discussion of legal principles by a Wisconsin lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like Avvo, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
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