I have tried telling my parents that the "attacks" happen mostly when I exercise and I still have breathing difficulties and chest tightness even after I have stopped exercising, but they keep claiming that it is normal.

if you're not overweight then you're probably right – it isn't normal to be experiencing chest tightness during/after exercising.
visit your doctor if your parents won't listen and go with someone you trust. if not, why not try a nurse a school (assuming your still at school) and explain what your feeling. It's better to know and get the right medication than wait for something horrible to happen (touch wood)
in the mean time, don't over work your body. if you get another attack, then sit down, calm yourself and try drinking small sips of water. I find that if you lie down, it makes it a teeny bit worse.
Hopefully you don't have asthma. good luck x

Exercise induced asthma is a real possibility. get your parents to take you to the doctor so he can write you a note for school and get you an inhaler. get them to take you to the doctor for a different reason – physical for school, birth control pills, make up something…

Go straight to a doctor / free clinic (if you're in the US), or talk to someone at your school who might be able to help you see a doctor despite your parents not wanting to take you, ie. a guidance counselor or a teacher that you think might be able to give you some advice.

id go with "iv had normal exercising breathing difficulties and what not but when i get one of these attacks its much worse, like im trying to breath though one of those coffee straws, and i know that much isn't normal"

you could:
-tell your teacher about it
-visit a doctor with your friends
-tell you parents again and stress on the negative impacts of it

Visit the doctor yourself.
then ask the doctor to tell your parent.

it is. the more you exercise the better it will get

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