I can't quite see the problem in referring to oneself as a cat or dog's Mummy or Daddy as long as you do not get completely unreal about it.
It's how you treat the pet that is the issue not what you call yourself.
I refer to myself as my budgies mummy.
What was the name of the corner shop you went to as a child?
What age did you reach your current height?
absent. Thinking of you.