In fact, it is not important whether the parents sign the message in the end, but mainly to express the expectations of all parents. As for the final signature, it is not particularly important. If it's required by the school, it's best to sign it, because it can distinguish which baby it's for. According to the situation, if it is just what parents want to say to their children, there is no need to sign. If it is required by the school, it depends on some specific requirements of the school.

Or it is ok to write a father or mother at the end.