If you get STD testing on your parent's insurance will they find out from the bill?

Your best bet at making sure your testing is 100% private is by finding a place where you can do anonymous STD testing. Rather than use your parent's insurance, I would just pay for it yourself. It's usually quite inexpensive and in some places it's even free. If you don't want anyone, especially your parents to know, I wouldn't use their insurance just to be safe.