I think your parents are absolutely right. There is no placating someone like this, and he will continue to wear you down. You aren't going to "save" your daughter from "contentiousness" if you are a nervous wreck.
I would suggest following your parents' advice to the LETTER.
And, in my experience, once you take a stand, and he's done posturing and playing games, it won't be as contentious as you feared.
And BTW, I'd be shocked if he actually spoke to a lawyer at all. It's just another ploy to get you to do what he wants.