BenZ is right - on ANY network, such detailed information (meaning actual content) is only available to the user, unless released under court order. I don't think your parents will be attempting to retain a court order for the purposes of spying on you, nor could they likely get one if they tried - not based solely on wanting to know what's going on in your life...pretty sure you're safe, unless you get into some legal trouble that involves your phone. That is a rather valid concern, though!
They cant see them online. But they can take your phone and physically check it! That is what we do with my stepdaughters. Personally if it isnt something you want your parents to see you should not be txting it! We check now because we have found inappropriate txts on her phone. So she knows we check it. I pay for the phone so feel it is ok to do this. I know not everyone will agree with it.