The stock in a custodial (UTMA) account already belongs to you, and always did. Your father was just managing it for you while you were a minor. He should have told the brokerage to put in an account of your own years ago, when you reached the age of majority in your state. It would be best to do this now. It might be more difficult or complicated after he passes away.
Since your father never owned the stock, you do not inherit it when he passes. It's already yours. That means there is no way to avoid tax on the gain if you sell it. Keep in mind that your gain would be long-term, and therefore tax at a lower rate.
You could hold on to it and leave it to someone in your will (or divide it up if, for example, you have more than one child). Someone who inherits it from you would get "stepped up basis." That means that the increase in value for the entire time that you owned it would escape tax. Your heir(s) would pay tax only on any increase in value from the date of your death until they sell it.
It might be possible to trick the system by giving the stock to one of your parents as a gift, then you inherit it when your parent passes. That would give you the stepped up basis. But I'm not sure of the legality of this, since it's an obvious tax avoidance scheme. If you think you would want to do this, you should consult a tax lawyer.
