Parents do not spy on every act of communication their teens perform. They do not spy on their conversations at school, and they do not generally listen in on their phone conversations with friends at home. As texting is a form of communication, teens should have the same right to privacy when it comes to texting.
We all desire a certain amount of privacy in life, and text messaging is simply THE form of modern communications between friends. As such, teens should expect that their private text messages will stay private. However, I also do not fault the parent who wants to keep tabs on their child, and specifically states that their text messages will not be private. At that point, the teen no longer has the expectation of privacy.
No, teens should not have the right to privacy from their parents when sending text messages. Their parents pay for almost everything in the house, and the teens are legally bound to the parents until age 18, in most states. Also, the parents pay the cellphone bills in most cases, so the parents have a right to know that their children are behaving as they want them to and that they are not committing crimes. On some plans, parents can simply log in to the carrier's site and look at the text messages that were sent on their teens' phones.
If the parents are footing the bill for the text messages, they should definitely have access to read what their teens are sending and receiving. But in any case, the parents should be able to read their teens' text messages to know who the teens are communicating with and what their teens are up to. Parents need to know that they can trust their teens to do the right thing, and they need to know that their teens are not abusing their text messaging privileges. They also need to be able to protect their teens from harassment and threats, and removing the shield of privacy helps the parents understand what their teens are enduring.