Not all kids are ready to be pet owners. Even though moms and dads are usually the ones who end up taking the most care of the family animals, it's important to make sure your kid is mature and safe enough to have a big pet before you get one. Before getting a dog or a cat, you might get them a lower-stakes animal like a fish or a gerbil to see how they handle it. Or, you could bring them around friends' dogs and cats to see if they can respect their boundaries. My parents waited until I was nine and my sister was seven before getting a real-life dog, and I think that was a safe bet.
Even if a kid has reached double digits, that doesn't necessarily mean that they're ready to have a pet. One woman promised her 10-year-old nephew he could have one of her dog's puppies, only to realize he was demonstrably unfit.