"She knows her parents-in-law's birthdays."

I'm with CK - it still doesn't look right. This sounds like a case for Prepositional Phrase!! More powerful than an apostrophe! Sexier than a dangling participle! Able to be strung together so that a sentence is right next door to incomprehensible! [coronet flourish] It's some words... it's inane! It's Prepositional Phrase!

"She knows the birthdays of her parents-in-law."

My work here is done.