I think that CPS is an awfully strong first step.
Asthma is a chronic disease that can only be managed and controlled. It will not go away.
Asthma is nothing to fool around with and untreated could, depending on severity, cause serious effects, and in extreme cases, death. It is nothing to try to handle on your own, whether @tinyfaery has been successful or not, is no indicator that you would be.
I think the best course would be to sit down with your folks at a time when you have put your thoughts together to present to them ahead of time and when you are feeling calm.
Here is a sample dialogue to give you a start. Put it into your words.
“Mom, dad, I respect your beliefs about medicine and the medical profession.
However, I feel differently about these issues. I would like you to recognize and support my choices for taking care of my health.
Not being able to breathe comfortably is a very scary feeling. I want to fill the prescription for the rescue inhaler for my asthma. It is important to relieve my anxiety that if I get in trouble, I can help myself.”
Your doctor probably also referred you to an asthma educator. That would be a very important session for you so you can learn about self-care, symptoms to watch for, how to chart your asthma triggers (time of day or night; any trigger from exercise, allergens that may cause an asthma attack for you, etc.
Tell your parents that getting the knowledge you need to manage your condition is important.
If they are really close-minded and do not respond well to your reasoned approach, my suggestion would be to enlist a clergyman or a teacher you trust to speak with your parents again. A school professional, if your parents do not respond, would be able to initiate a Child Protective Services inquiry on your behalf without you being implicated by your parents if it were to come to that.
I am so sorry this is happening to you. Please take care of yourself. Keep us posted on what happens.