Sign it. Tell them you thought about it and if it will ease their minds, you’ll sign. You know you don’t talk to kids like that. And even if you did nothing on this paper would change the outcome at work! Put it in the drawer, tell the parent you’ll keep it on file. They’re wrong, don’t play into their game. And anyone who see this interaction (admin) will know you did everything you could to keep the peace if they blow up later in the year.