If you keep bothering them and irritating them again and again, then these tips will prove useless for you. Just like you hate it when they keep nagging you, the feeling is mutual for them too. Here are some methods that you can try to make them see your point of view so that they agree to get you a dog.
Show that You are Responsible
Don't just say it, prove it. They might have heard it repeatedly from you, and yet they needed to tell you to do your stuff on time. So, telling them to get you a dog saying that you will be responsible won't work. Prove it before you get the dog. Get the idea of a dog into their head, tell them that you will be responsible and start working on it.
Clean your room, help your dad wash the car, help your mom in the kitchen, lay the table without being asked, eat your food properly, and so on, by yourself. You might think they aren't noticing, but they are. They might not be convinced by your changed behavior for a few weeks, but do not give up and they will soon be convinced. Patience is all you need.
Do your Research
As you demand a reason when your parents ask you to not do something, they have the right to ask you why you need the dog, so be ready with a really convincing answer. Tell them how you hate to return from school to a lonely house, how you will reduce time on TV shows and go to the park, and play with it or train it and how it will be good for your health.
Don't forget to focus on points like how dogs provide security from robbers by guarding the house at night, how they make loyal friends and keep their masters happy, etc. If you tell them something like, 'Because John has it..", then you can be sure that it won't work.
Show how Serious you Are
To convince your parents to get a dog, prove to them that you are dead serious about it. Do your homework really well, research about the breed you want, from where you can get it and its cost, its temperament, the nearest vet, various shots that you need to give your dog, how to take care of it, and also, find out about the best dogs for children, etc.
A Second Dog
The earlier mentioned tips were mainly for those who do not have a dog, but really want one. However, there are some kids who, despite having a dog demand for another one and their parents, though convinced the first time, have hard time agreeing the second time. Therefore, to convince them, tell them about how lonely your dog is and craves partner/company.
Show that you can manage another dog by bring more responsible towards the first one, tell them that the training and caring experience with one will help in managing another dog. These explanations will help convince them. However, if you have been irresponsible despite them getting convinced the first time, then getting a second one will be difficult.
Beg, Plead, and Shed Tears
This is the last resort and this step is to be used when everything else fails. Just get down on your knees, fold your hands, shed a few tears, and plead with your parents. Give them a 'puppy' look and try to melt their heart and change their mind. If your neighborhood is not good enough for a big dog, go for a small one, just to please them.
These pointers will prove helpful if you are serious about getting a dog. And please do not keep bothering your parents again and again and start with all these 'dog wanting' strategies as soon as they come back from work or when they are in a bad mood.
Choose a time when they are relaxed and in a good mood and then pop the question. Be polite and DO NOT create a scene if they say no for the first, second, or third time. Let them ponder over it for a while before they come to a conclusion. The end result will be great.