Do your parents believe that pushing you to get a job will help you move beyond your phobias? These things are not easily understood by people who do not have them.

And education is expensive. Most students have to work part time while going to school to help pay for their basic needs, like shampoo and mac & cheese.

You stated that you feel you have serious social anxiety. Something like that will have a big impact on your success at school as well. If you are too socially anxious to have a part time job, how will function away at school?
You may have confused your parents by using that as a reason to not work. It comes across as simply a bad excuse.

I hope you and your parents can work through this together.