what can i do to prove my parents that i don't have an eating disorder
Hello, and thank you for your question. This is a difficult situation, where your parents' disbelief in your recovery is going to deprive you of an opportunity to travel. Your parents' reaction indicates a pattern of them thinking that you are the cause of their problems and that you should solve the problems. I don't know your entire history, but this dynamic likely played a part in your use of bulimia as a coping mechanism.
A good first step at this time is to invite your parents to go to a medical appointment with you where you can have a physician evaluate what you ate/what happened, and hopefully support your assertion that you were genuinely sick and vomited because of a physiological issue, and not an emotional or psychological issue. A professional opinion can go a long way to support your cause.
Your parents' blow-up is indicative of some things that have been left unresolved; an underlying pattern for Bulimia. Bulimia is not created in a vacuum. Families tend to have an identified scapegoat in the family that the family members can project their shortcomings and issues onto. This allows family members to live in comfort; not look at themselves and their contribution to whatever problems they are having, and they do not have to work on themselves. It's quite possible that that is what happened in your family. You stated that your father started to accuse you; told you that you are only there to torture them; that you made their life miserable; and he said that he'll never believe you again. It's easier for the parent to convince themselves that the child made the parent's life miserable, rather than for the parent to look at the choices he/she made and take responsibility for his/her life. It's not surprising that the family's punishment is to not let you travel abroad. If you suddenly aren't there to be the scapegoat that absolves them of their responsibilities or distracts them from their issues, they will be forced to look at themselves as the cause of their own problems.
When you received treatment in the past, were there any family sessions? Your family would benefit from family therapy. They need to understand that there is no singular Bulimic family member in the family. The family is Bulimic because each family member contributes to the manifestation of Bulimia in you.
To sum it up, see a physician with the family, and ask the family to enter family therapy with you. I wish you the best.