I think your parents are just trying to be helpful. So little is known about this problem even by dermatologists.
Products called 'natural' may contain any number of synthetic chemicals which can cause irritation.
Watch to see if there is any change in the severity and frequency of your EC after stopping. If you notice a change within 48 hours then you know that the toothpaste may not agree with you. Do not expect the peeling to stop entirely in this time. If you notice no change at all then the toothpaste has nothing to do with it so you will need to keep researching here.
If you notice your EC improve post back here and I'll see what else I can offer for advice.
p.s. you should most definately see about that patch test, I have had one. Biopsys though are a lot more intrusive and best avoided as they will seldom give any answer to what is causing the EC, some on here have had biopsys and regretted them.