It’s tough to get rid of dandelions in a completely pet-safe manner. They often grow bang in the middle of your lawn. Ordinarily, you would use a selective herbicide capable of decimating the dandelions without damaging your grass.
Unfortunately, these products are not pet-friendly. You have two options: either hand-pull the dandelions or use a pre-emergent herbicide like cornmeal gluten. This will kill dandelions effectively.