There is a difference between running and running lots of miles to train for a marathon.
If you do it - come up with a good plan to not ober due it.
Maybe you can take this in steps and meet your parents in the middle. This year train for 5k and 10k races slowly building up your mileage and see how it affects your body.
I would not hide it from parents - Looks for some good articles on the benefits to teens and benefits to your asthma. Then work together to find a training program that you both can be happy with.
Once you are able to take the 1st step, they will most likely be more willing to let you make the 2nd step in adding mileage and intensity.
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