Messages on phone bill

The bill will not display the contents of messages. iMessages (blue bubble) are only data transfers and will not display on the bill. SMS, text messages between an iPhone and non-iOS device will display the number texted, but nothing about the content of the conversation.

If the folks have your Apple ID and password and have authorized another device, they will see the messages. That’s no different than you possessing and using several devices. You can see all other devices logged into your Apple ID from iCloud..com. Outside of your own Apple ID, no, messages are not shared. Not unless the folks are involved in the particular discussion.

Best to avoid those sorts of discussions in general, though. Parents can and do have other forms of leverage.

As for billing, and on whether or not your contract includes some quantity of SMS messages or not, SMS messages past what’s in the cellular plan (if any) can incur carrier charges, and those charges do show up on bills. That’s a charge for the SMS messages themselves, and not the message contents. These SMS messages are the “green” messages shown in Apple Messages. The Apple messages in Messages that are part of your data plan are shown in blue.