can iMessage be traced on Verizon phone bill?

You're welcome. What I provided is public knowledge, but it sounds as if you have other issues going on with your parents that you need to resolve. That is up to you to handle.

iMessage cannot be traced on the phone bill. It is data, just like an email, and all it would show is a certain amount of data usage, no phone number associated with it.

iMessages are simply data and not identified as text messages on your parents phone bill...

You could get your own iPhone & account and not have to worry about hiding what you are doing...