I think most parents are keen to see their children engage in activities that help them grow their knowledge and show that they can take on responsibilities and follow through with the obligations that come along.
I would guess that your current behaviors will influence how your parents will react. Do you help with things around the house (regularly clean your room, take care of your brothers/sisters, respond when you are asked to help, etc)? If the answer is yes you will be in a better position.
As to how to approach your parents I suggest that you MAKE A PLAN and write it down.
- Where will the birds be kept (both coop and outside run)? How will those spaces be built?
- List the various work tasks that will be needed to take care of the birds and indicate how you will be able to carry out those tasks on a daily, weekly, etc. basis.
- Try your best to figure out how much money it will cost to get started and then keep going over the months and years that the birds will live.
- List the benefits that will come from having chickens (eggs, amusement, fertilizer for the garden, etc.).
- Figure out realistically how long it will take to get started.
- ??? other things you may think of.
Most parents will not expect a teen to do everything needed to undertake all of the things in your plan. If the plan makes sense, if they see that you are serious, if the family circumstances (where you live, other pets, available money, etc) make it possible you stand a much better chance of convincing them to help you in getting and looking after a small flock of chickens. If there is something that prevents them from saying yes, figure out what that is and what you can do to diminish it's importance.
This forum has many members that would love to help you to get started. If you have questions just ask (as you have already done) and you will get great suggestions. They will even help you with your plan once you get some basics written down. Keep asking.
Best of luck!!!