Right, pkd! Another tactic would be for your parents to approach the relatives from the standpoint of the benefit TO THE CHILD to be gained from interaction with animals, such as the family dog. When I was an infant (back before the earth had fully cooled!), my parents had no dog. So my parents got me one -- a little black Cocker puppy named Betsy. My dad brought her home and put her in my crib. Thereafter, Betsy and I were inseparable playmates during my childhood. When I was old enough, I was assigned some responsibilities in caring for Betsy, such as feeding her and making sure she had water. So, having a "canine buddy" is really good training for a child.
Having a child in close contact with a dog really presents no health risk to the child -- or to the dog. One of the many reasons dogs became popular as house pets is that they have almost no diseases in common with humans. Rabies is an exception, but it's a legal requirement that all dogs be vaccinated against rabies in every area I know of.
Thanks,
Richard