How do parents access their children's text messages on iphone?

The reason I ask, is because somehow my parents not only read my private text messages, but were able to get copies of particular ones sent to their phones. At first I thought it was a spyware app my Father somehow installed. So, I have since updated my IOS to a new version to hopefully have gotten rid of(or disabled)any spyware.

Until u actually over 18 and moved OUT paying for everything, U CHILD have NO privacy Legally.
B smart don't do STUPID.

They can have an app installed on your phone for that. But, they most likely go to the cell carrier store and ask for them. Employees of the cell carrier store can do that. But, this should all end once you become 18 years old as its illegal for your parents to do that without a good reason or your permission.

If you are under 18, well, you can't really do anything about it.

Some carriers may have features that let them see that from home but i know for sure Verizon and AT&T does not have that as you can only see what time texts were sent and received on that device.

Is it possible you are sending "improper" messages you do not want your parents to read? And who is PAYING for the phone service? The person paying has certain rights to make sure the phone IN THEIR NAME is not being mis-used.

They start by paying the bill on the account.

Depend on the phones, system, plans, ect. That you have. Some can request all transcripts or receive them monthly, some are spyware as you suggested, some can simply be requested like a bank statement from one date to another. There are lots of things out there, we have accesses to our kids phones, if we pick them up and our accsess has been removed, so is the phone. Pretty simple actually. It is a parents job to be aware, especially if there's a suspected issue. Better to have the kids pissed then the cops banging in the middle of the night.