If they are good pet owners, chances are they will make responsible adults and even caring parents. While no studies are really possible, it makes sense to assume that the opposite is true as well. Caring for pets at a young age certainly must be a good indicator of good parenting skills to come. Certainly no easy answer exists.
Kids are still kids.
They can only absorb so much in the course of their busy days. That is why when parents ask me what type of pet they should get, I always ask them what type of pet THEY want, not their children. Ultimately the parents are the ones left with the majority of the responsibility for the pets. Do not get a dog for the kids if you would rather have a cat. Don't get a puppy if you, as the parent, do not have the time for it.