You can probably start the process, if the power specifically grants this authority, but he will most likely need to be the one to make the final signature. Loss of parental rights is pretty serious and not to be taken lightly.
Every legal matter is fact specific, and there are often nuances in every case. This is intended for comment only, and does not create an attorney client relationship.
Terminating a person's rights is so serious that another person just cannot sign for that person. The person must sign himself in front of a notary.
Normally a judge appoints a person to represent the child's interest and that attorney normally interviews the parent terminating his/her rights to make sure that they fully understand what it means to permanently terminate their parental rights.
Harris County judges take a termination very seriously. Many years ago a superior court over-turned some terminations and re-instated some parents parental rights after the courts held that the terminations were not properly done. You can imagine how the children and the families were impacted after the parental rights were re-instated years later! No judge ever wants that to happen again so they take these terminations very seriously now. You might think the judge is being "silly" but they are actually protecting the child.
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He will need to be involved at some point unless he is incapacitated. Please note that terminating his rights does not necessarily result in him not being responsible for child support. The Judge can still order that he provide for the child financially. Also, a Court is typically not going to want to terminate parental rights (assuming there is not abuse or neglect) without knowing there is another adult ready to come in and adopt that child.
A power of attorney should only be utilized when the principal is unable to perform the act themselves. Unless your husband is incapacitated or out o the contry for an extended period of time, I would not suggest you to initiate these proceedings on his behalf by using your power of attorney.