Would it be appropriate to communicate with a friend's family about her eating disorder?

You should confront her about it first. Keep her accountable, be there for her. Call her often and make sure she continues therapy and eating. By telling her parents you violate her trust, and it sounds like she really needs someone to be there for her. If she continues to get worse, and stops going to counseling, than as a last resort go to her parents. I think the best thing is to be open and honest with her, tell her that you are worried about her and that you want her to be healthy. Generally individuals who suffer with eating disorders have other issues that they are dealing with or have dealt with (such as ual abuse). If she has been ually abused/aulted going to her parents wouldn't be helpful simply because individuals who have been ually aulted need to be put in situations where they are put in control. Like I said approach her and come up with a solution together, instead of deciding one for her.