Parents, or anyone else for that matter, do not have access to the content of your text messages unless they have your phone in hand or a court order for Verizon to access them. They can view the billing records to see the numbers that were involved in texting.
Your parents have every right to know what happens on the phone, as they are responsible for the bill, and for your actions until you reach 18. Simple answer - don't do anything you don't want them to see.