The answer is "no." If you petition to terminate your own parental rights, your petition will be denied. However, if the custodial parent is agreeable to terminating your parental rights, you could coordinate and cooperate on this. There are some key aspects of this scenario of which you have to be aware.
The custodial parent and I have agreed to terminate my parental rights - now what?
First, the custodial parent will need to file the Petition to Terminate Parental Rights. The parent whose parental rights are being terminated can consent to the termination if it is in writing, witnessed by two people, and everyone's signatures are notarized. These usually also have a waiver of notice in them, so in practical terms, that would be the end of the process for the non-custodial parent. However, you would want to know that your parental rights were, in fact, terminated so you know you are no longer responsible for paying child support.
If the custodial parent and I agree to terminate my parental rights, is it for sure going to be granted?
Again, the answer is "no." There are various considerations for the Court. Is there someone ready to step in and adopt the child? If not, then the District Attorney, child support enforcement division has to be notified of the proceedings so they may object to the termination of parental rights, on the basis that if your parental rights are terminated, they no longer have a child support paying parent to chase in the event the custodial parent goes on government assistance. They do not like that, and will rail against it. It is possible to terminate even without a person ready to adopt, but the custodial parent will have to convince the Court that they have never been on government aid, and it is highly unlikely they ever will be. Obviously, with an adoption pending, the District Attorney doesn't need to be involved.
So everyone - including the Court - is on board with the termination... I no longer have to pay child support once the Order is entered, right?
Not exactly - your prospective child support obligation (your obligation from the point the Order terminating your parental rights is entered by the Court Clerk and it is file-stamped) goes away, but if you have child support arrears, those are not extinguished unless there is an agreement to extinguish them between the parents, and even then, not always...
What can I do to extinguish child support arrears?
If ALL of the child support you owe is owed SOLELY to the other parent, it can be extinguished by agreement between the two of you. However, the custodial parent does not have the right to waive child support due to the District Attorney, child support enforcement division as a result of seeking reimbursement for government aid given to the custodial parent. Only the DA can waive that, and they don't. So it is important to know whether there are any arrears owed to the DA. Custodial parents aren't always forthright about this, just so you know, so you should personally check with the child support division. Also, even if the custodial parent extinguishes arrears and the DA was not notified and the DA is owed some arrears, the waiver of child support won't be effective on the DA.
Terminating your own parental rights is tricky, but can be accomplished, as long as everyone is on board. There are pitfalls, and it is always a good idea to seek legal counsel before taking such extraordinary steps, since termination of parental rights is permanent and cannot be undone.