I guess it would depend on your "ex-relationship" and how things ended? I've never been married or had kids with anyone except my husband, but I am friends with 7 guys that I dated (2 were serious millions of years ago, both are good friends, one lives in Spain and the other in Russia now). I am NOT friends with one that has no exgfs as a friend (he's newly married, and I totally respect that; we get random updates from our moms who bump into each other from time to time), and I am NOT friends with one guy who makes my stomach ill just thinking he still exists (ha).
My old boyfriends were just that: boys, who were friends (and more), but it was a million years ago and we still can laugh at the same public posts/jokes, brag about children, etc. It is NEVER talked about by any of us that we were bf/gfs because we all are married with kids and there's no reason to go there. I have them in the "acquaintance" or "high school" folders (depending on where they belong), but only 2 are in my "close friends" folder. I suppose with those 2, we're all still "good guys" to each other because relationships ended due to moving away, not things going sour. I don't talk to any of them often though, with the exception of following a winning racehorse he owns. I can't imagine my mom and dad being friends on FB especially since my brother and I are grown. Their marriage ended badly, and it wouldn't be a good idea. (I also don't comment publicly on anything my dad posts; just in private messages, to spare my mom the annoyance). Yay, FB drama. :P