I've had something similar happen to me.
I didn't reply to the parent, although I admit I was sorely tempted to - it's not easy to sit and read the character assassinations and contempt of your own colleagues and not do a thing about it when you know it's all based on lies. Especially when they start on about "I would never do this". I'm often surprised at how quick other childminders are to accept that such unprofessional behaviour is apparently so commonplace they feel the need to assure "We're not all like that". Thanks to my experience, I'm much more inclined to believe that very few of us are "like that".
It was obvious that this parent was simply trying to get out of paying the notice period and I had previous reviews, emails and other documentation where the parent had sung my praises to the sky about how wonderful I was, how she couldn't do without me etc. etc.
I didn't ask for the posts to be removed although, if it ever happened again, I would do so. However, I didn't feel I was easily identifiable. Maybe I would have reacted differently otherwise. My reputation means a great deal to me.
When the parent acted on the poor advice given, I tried to stay calm and professional, corrected the out of date advice (she'd been told to report me to the Council, for heaven's sake!), and reported the whole situation myself to our Early Years department. Much easier after that: they fully supported me and advised me how to proceed. Early Years and OFSTED are well used to parents trying to wriggle out of notice periods. It's one of the reasons OFSTED absolutely refuse to get involved in any contract dispute.
I got my notice payment and the parent was warned by Early Years against slander.
I don't think we can claim a breach of confidentiality by parents, but then neither can the parent since he or she has chosen to make public their grievances.
Perhaps we ought to think about how we respond to these posts in the future. Maybe referring them to the NCMA, who are there for parents as well as childminders, or some similar body. Perhaps we shouldn't be condemning parents or childminders without knowing all the facts from both sides. Definately, if we're going to advise, we should be certain that the advice is good, not poor, outdated advice that misleads parents.