The title may not be entirely correct..

Is there a word or phrase that describes a mother/father who is overly anxious about their baby's health? 

Not a word to describe both parents together, a word to describe one parent. 

Also, a word that could be used in describing the parent to others as well as directly to them. Something like *'Worrying Wendy, Delirious Dad, Medicine Mad Mummy'*. I suppose they must have coined a phrase for this in the medical profession.