I can just imagine your frustration with getting through to your parents and dealing with your health issues as well.
It is not easy for parents to let go of their children and stop laying down the law. You are expecting your parents to do that earlier than normal in the sense that technically you are still a child and they want to still be prescribing parents, while in fact your dis-ease has caused you to take on responsibility for yourself much earlier than you normally would have.
It feels like in your situation you will need to take even more responsibility by helping your parents to cope with the situation. How can you do that?
You will have to set specific and tangible goals for yourself in terms of school - not so much how often you can attend, but rather what outcome you want to see in terms of achievements or academic results. And these results do not have to be top of the class - you know what you are capable of. Then you need to sit your parents down and explain to them what you want to achieve at school. This will give them peace of mind that you are not just shying away from school, but rather than you want to achieve what they want for you despite the odds.
I would suspect that they feel much anxiety about not wanting you to miss school, and if you can share your goals with them, they will feel less anxious and out of control, and more willing to trust and support you. Of course once you have set your goals, you will have to make a real effort to achieve those goals despite your dis-ease.
When they understand that you are serious about school and that you also want to manage the situation as best you can, they may be much more supportive on your "off" days so that you can feel stronger and achieve more with their support on your "on" days.
There is also something else you can do for yourself.
You need to get to the bottom of what set off this vomiting in the first place. You say it came out of the blue, but I always look at a deeper level to see what happened before the symptoms started. And no, I am not recommending that you go for expensive therapy and contemplate your navel or wreck your brains trying to remember what happened at the time.
I want to recommend a far more powerful and inexpensive method of getting to the bottom of this, namely writing a journal.
This will take courage and discipline, but I would not be surprised of you scratch open the real cause - the real stuff that you ‘cannot stomach’ - and also a way to deal with it all.