Lorelai’s mistreatment of Rory’s childhood pet hamster is Rory’s reason to not allow her to get a dog, a nice reversal of the usual parent-child relationship.
It’s interesting that Rory says that Lorelai doesn’t need a dog as she has her, suggesting that Rory sees herself as more of a pet than a daughter. It is only when Lorelai no longer has Rory that she gets a dog – although she treats it more like her child than her pet.
It is not known when Rory might have had a pet hamster. They didn’t get their own house until Rory was eleven, suggesting it might have been after that. However, by that stage Rory should have been old enough to care for her own hamster, so that Lorelai wouldn’t need to do anything. If Rory had a hamster during the mysterious “lost years” between perhaps four and ten, we don’t know what circumstances Lorelai and Rory lived in, but whatever it was, it apparently allowed pets.