From your Parents' Viewpoint
It is your parents job to provide you with food, shelter, and clothing. It is not their job to provide you with a pet. A pet is a lifetime commitment that all too often is dumped on the parents who did not want that responsibility in the first place. A pet is an expensive purchase, because it has on going costs, such as food, veterinary expenses, and ongoing care.
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From the View of the Pet
A pet requires a lot of time. They must be tended to daily, fed, watered, cleaned, and exercised. Some more intelligent animals also require mental stimulation to prevent them from being bored. An animal isn't something you get to make yourself more cool, that animals life, and happiness is in your hands. One of the WORST things that can happen to a pet is that a kid gets it after begging mom and dad, but years, or even months, later the kid is either bored or moves out and cannot take the pet with them, and mom and dad do not want it. Many pets are abandoned in shelters or euthanized because children who once pleaded for a pet are no longer able or willing to care for it. In all fairness to an animal, unless EVERYONE in the household wants a pet, you should not get one. Unless your parents are fully willing to help financially and understand the lifespan of the animal may exceed your ability to keep it, then you should not get a pet. I am sure you think you will always be willing, and able, to look after your pet, but in reality this doesn't always happen.
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Research
One thing that will help you is knowing what kind of pet you want and what kind of care or expense is required. Do not be set on one kind of animal. For instance, many kids want hamsters but find that the hamsters habit of running on the wheel at night to be annoying, as such a small rabbit would have been a better choice. Maybe you want a Border Collie (a dog used for herding sheep), but, if you do not have time to keep a Border Collies mind stimulated it will become bored and destructive, so a Shih Tzu, or cat, might make a better pet.
Write down the pros and cons of every pet you might consider. Include things like lifespan, size, activity level required, cost, and housing equipment needed. If it is a horse you are asking for, you must realize these have huge ongoing costs, especially if you must board them at a stable. Do not even ask for a horse unless your parents are horsey people and have a good income.
You will need to check if you can have a pet. If your parents own a house and you want a dog, you should have a fully fenced yard or dog run. Chained dogs have a tendency to become aggressive. If you rent you must make sure you have landlord approval for any pet. If you are planning on moving, you should wait until you move before getting a pet or you may find you must give it up, as very few places allow pets.
If you are looking into something more exotic, like a Sugar Glider, Ferret, or Rat, you need to check if they are legal as pets in your area. Remember most exotic pets require more specialized care and food.
Things You Can Do
Try being more helpful around the house to demonstrate your level of responsibility. Keep your room clean and help with other areas too. Get a part time, or summer, job so you can help pay some of the bills your pet will have. Save your money for the pet and supplies, rather than spending it on other things.
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Things You Cannot Do
Do not beg. It might work, but it is the wrong way to get a pet. Your parents will never feel good about giving in to your constant begging, and will hold it against the pet. They will just want to "be free" of the burden, a constant reminder of giving in. Do not guilt them, or in anyway try to manipulate them, into getting a pet if they really do not want one. In this case it may be best to wait until you are older and can get your own pet after you move out. Generally, the more you beg, the more a parent will resist.
Do not lie, do not tell your parents a certain kind of animal is easier to care for than it actually is, or that it is not "smelly", for example.
The Decision
As mentioned, the decision lies with your parents, it is them who will ultimately be the ones who make the commitment. They should have final say in what pet you can or cannot get. They should be able to select the pet, this will help them bond with it too. Most places will not sell or adopt to children, and rightfully so. If your parents are getting you a pet and are saying "If you do not look after it, we get rid of it." then by all means DO NOT get a pet. If they show no interest in animals then DO NOT get a pet. The pet should be referred to as a family pet, not specifically "yours". After all, you might mean well now, but what happens when you move out, or go to college, and cannot find a place that allows pets? If your parents made it clear they do not want the responsibility, then you can be sure they will not keep the animal, and older pets generally do not find new homes.
Success!
If your parents agreed, I am sure you are thrilled and now have to select the pet. Again, it is critical they play a major part in the selection process. The best place to look for pets is your local animal shelter, they generally have the best value, and selection. Since most are non-profit, they are not motivated by money, so will be honest about each animal. You may find that you have to return several times before finding the animal that is right for your family. One of the benefits of adopting from a shelter is they provide a guarantee and have lots of information available, often free of charge, plus you are truly saving a life, rather than supporting an industry of breeding animals solely for profit.
Getting pets "free" out of the newspaper is generally more expensive because they may not have had any veterinarian care, such as vaccinations or worming. Sometimes older pets are offered for free, and these may be worth looking into, especially if they are already trained, vaccinated, and come with supplies.
The other option is to buy from a breeder. This is best if you are looking for a purebred, registered, animal. Reputable breeders breed with the main interest of improving the breed and have shown the parents to prove their worth as breeding animals. Finding a reputable breeder is a whole other matter, you can ask a vet, dog groomer, or your local dog club (these will also be able to tell you where your local shelter or SPCA).
Now, you must remember to stick to your promises to your parents and to the pet.
