Our niece lives with us and won't stop from convincing my parents to get her a dog, my dad is still recovering from a surgery and getting a dog might interfere with his recovery. How do I tell her to stop?

I guess you can buy her a substitute instead? A stuffed toy dog, maybe? Or anything that can keep her distracted from wanting an actual pet dog. You can discuss to her as well that she can get a pet when your dad has finally recovered.