You’re an incredible, kind-hearted, and very thoughtful partner. Kudos to you for doing your best to help your girlfriend.
I don’t believe it is overstepping your boundaries at all to bring this to her family’s attention. But I believe you should ask your girlfriend first before bringing the matter up to her parents.
Any remark, no matter how “trivial” or unintentional, can seriously harm someone struggling with an eating disorder. Every word can be dangerous or triggering, and her parents must understand this in order to help her beat the disorder.
If they don’t understand how triggering and insensitive their comments can be, they will continue to make them and therefore continue to reduce your girlfriend’s hopes for recovery.
I would definitely bring all this to her family’s attention, but of course you need to do what you and your girlfriend think is best.
Good luck to you and your girlfriend! Best wishes.