ask her to get you a small pet and be responibal with ans maybe she will. or even better ask her if you can volinteir to take care of your neighbor's dog why shes away or sick and do a good job and maybe she will relize your recponcibal.
mlynn101 made a good point about taking care of a friend's or neighbour's dog to prove that you can do it. It'll also be good experience for you. In fact, I recommend that over the option of getting another pet first. Different species require different types of attention so if you specifically have experience with dogs then that'll be great!
Another important thing to be aware of when thinking about getting a dog is how much time they require. You should make sure you will have enough time to do everything you need for it; it's worth doing some research into the breed of dog you're thinking about and finding out exactly how they need to be looked after (as some breeds require more attention than others).
Lastly (and this is important when adopting any animal) you need to be sure that you won't get bored with a pet. For some, owning a pet is a novelty at first, but eventually they lose interest and the animal is then not properly looked after. This is where previous experience with owning animals is handy, just so you know that for you, looking after a creature is a lifestyle and not just a pastime you will get bored with.
