Take some initiative and show your father that this is important to you and that you can be responsible enough to care for a pet.
Make up a written proposal (it doesn't need to be very long) This will show your father that you have put thought into it and that it is important to you.
In your proposal you should put that you are willing to show how responsible you will be with your pet by offering "sitting" services for pets of your family's friends. (This can also give you some pocket money to spend on the pet)
You will have to follow through and be responsible though. Make sure you give the pet food and water. Clean up after all it's messes. Give it exercise.
While you are writing your proposal - ask your parents questions to find out if they had pets as kids. If they did, ask what was the best and worst things about having a pet. If they don't want pets, try to find out what the reasons are. Are there allergies? Afraid of the mess? They think they will be the ones doing all the work? Don't want to spend the money?
Pets can be expensive. There is the initial cost of the animal. Then there is the constant cost of the food, vet bills, there may be registration/tag fees every year in your area if dog.
If they don't want to spend the money. You will need to think it out and put in your proposal the expected costs (include all of them to show you understand the problem and are mature enough to face reality) and show how you expect to earn the money and how long you expect it to take.
If you show that you have put thought into it, that you are determined and that you are willing to work for it, I bet your father will be impressed and willing to help you get to your goal of having a pet.