Sometimes parents don't realize how mature their children are and often need persuading to extend curfews or allow their child to date. Being able to persuade parents to do something for you isn't as hard as it seems. The trick is being able to approach them in the right manner.
Research the subject you are trying to persuade them about; for example, if you want them to let you have a dog research proper dog care and dog training. The more information you have on the subject the better you look because it doesn't seem like you are making decisions on a whim.
Approach your parents in a mature manner and tell them you have something important to discuss with them. Explain your point of view and give reasons for why you feel that way.
Listen to their side of the argument. Explain to them that although they may feel like they are right you have done plenty of research and feel like you are ready to take on the responsibility.
Tell them they don't have to answer you right away. Give them time to process what you have told them and by giving them time to think you look like a mature person that is able to make her own decisions.