A year and a half ago they signed an agreement agreeing to joint custody and giving Bristol primary (not sole) physical custody, with Levi having visitation rights twice a week. You can read the actual document here.
Termination of parental rights is usually obtained when a couple wants to give their child up for adoption. Each parent has to sign away their parental rights, which means under the law they cease to be the child's parent. They get no visitation, no say in how the child is raised, there's no taking the kid to see his family, etc. It's permanent, it's final, and it means the daddy isn't the daddy anymore. Judges are very reluctant to grant a parental termination without the agreement of the other party, unless some serious physical or psychological abuse has taken place. We've heard none of that. We've only heard Levi tell of how much he loves his son and how difficult the Palins make it to see him.
Memo to Bristol: All divorces hit bumps and roads. You both have to compromise. There's no reason Levi can't still be a father to your child, even if you marry someone else and your child gets a step-father in addition to having Levi as his real father. Your plan is cold-hearted and unfair and in no-one's best interest but possibly your own..
What you should do instead of twisting Levi's arm to do something he doesn't want to do (and very few other men would agree to do) is go back and petition the court to modify the agreement by allowing you and your son to move to AZ, and making it up to Levi by increased blocks of visitation. You can plead your case, he can plead his, and the Judge will decide.
Courts are concerned with the best welfare of the child, not satisfying parental whims such as moving across the country. So you might lose. Maybe your lawyer has already told you that you will lose. Many states won't allow a divorced parent with a young child to move out of state without the permission of the other parent unless there's a compelling reason. It's not in the best interests of the child that a child be deprived of his other parent. There's no need to ask for such a drastic measure as termination of parental rights.
If parental rights are terminated, how are you going to tell your son he no longer has a father? Are you going to tell him, "It's all right, I found you a better one?" The child is going to feel abandoned by Levi. Are you going to lie and say Levi didn't want to be his father any more, or wasn't a good father? Your custody agreement prohibits both of you from speaking ill about the other.
Grow up, Bristol. You had a child with Levi and he's tried to have as much presence in the child's life as you will allow, which isn't much. You don't get to kick him to the curb. I'm sure there are broadcasting schools or junior colleges in Alaska. If that's your chosen career path, you may just have to sacrifice (a word that may not yet be in your vocabulary) and pursue your career in Alaska, so Levi can exercise his right to visit with his child. Or you can agree to let the child visit Levi in Alaska (or wherever else he is living)for 3 to 5 months a year. Suck it up. Life isn't always an ice cream cone.
Assuming the judges in Alaska aren't all in your mama's pocketbook, you may just find if you push this too hard, you will be the one who ends up with block visits in AZ while Levi ends up with primary custody in Alaska. It's happened before.
Selfish, selfish Bristol, in my view. Children are born with two parents and they deserve to grow up knowing both.