That kind of minor surgery can be cured, and adults will feel that it is good to be cured, and accept it slowly. You should persuade your boyfriend not to have psychological pressure. So you really can't give up on him. Rabbit lips are not easily accepted by ordinary people. It's good for you to understand your parents. You should let your parents see your persistence, let them see that he is very capable, loves you very much, and has.
You will be tired before your parents agree, but his family will help you through it. You must let him keep confidence in you, use the right method and persevere.