Hi, Lewis!
It's informal American English. (Maybe British, too. I'll let one of them confirm or deny.)
When we say "Don't get me started," it means that there is a lot we could say on the subject - more than you want to hear.
To rearrange it a little: "If I get started talking about those classmates, I'll lose my temper and rant. So do me a favor and don't get me started. Trust me, you don't want to listen to what would say."
The phrase "Don't even" is also tricky. It's highly idiomatic. Basically, it's a more forceful way of saying "don't." The parent could have said "Don't get me started," but chose "Don't even get me started" for added emphasis.
Good question. I hope that helps.

English Teacher
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