I'm really sad to hear about your problem. It really irks me when I hear about closed-minded health professionals. But they are not all like this, I know its a bit of a hassle but you should look for a good therapist rather than a psychiatrist, especially someone who is open-minded and well acredited. I work with doctors and medical professors and I see it all the time
Anyway, I'm quite open minded about alternative therapies but the reason many doctors or academics frown upon these therapies is because there are no research studies done to show any correlations (for most anyway) or long term effects. For example, the link you provided Airana, the first thing an academic would say would be, "there are no studies linked in the article, just a few years and doctors are supposedly quoted, how can I verify this?". The article is biased, no matter how true and relevant the information actually may be.
I do believe that food has an important part to play in mood disorders (depression, anxiety etc.) Asperger's however is related to cognition and development therefore your health care team and parents are more apprehensive about you trying any alternative therapies (probably) because its basis is more neurological than say, depression. Maybe try to persuade your parents to see a naturopath, especially if you want to stop relying on psych meds like. Talk to them honestly, as best you can, about your dilemma. As far as medication + fasting, I would strongly recommend you don't do it.
edit: also have you been given speech, language, social etc. therapy for AS?