new born allergic?

Hey guys. I'm a little nervous about my pets. My parents putting fear in me bout my house dog and cat. I have my baby Wednesday and they are saying that I don't need my baby around my pets because it could be allergic. Baby can get sick. My cat is a inside cat. It doesn't know a thing about being outside. And my dog either. He goes outside to use it but other than that.. they are completely trained and house hold pets. Now our pets only stay in our living room and the kitchen.. put hallway has a installed pet gate.  we have a pet gate where our pets doesn't go to the bathrooms or bedrooms. They have never been back there. They sleep in living room on their pet beds and get along great. It's a pet free zone. But  Has anyone brought their newborn home and their baby has gotten sick because of their animals? What did you have to do? How did you handle having a newborn and your pets? I am praying u don't have to give away my pets.