You cannot monitor your child's text messages except by viewing them on his or her phone. Verizon cannot offer the service, because no carrier can read the messages that they handle. Apple cannot either, for the same reason. There are no legitimate apps that can do it either. And what messages would you like to monitor? SMS/MMS? Whatsapp? Facebook Messenger? Signal? Google Duo?, Telegram? Instagram? pinterest? The only possible way is to install spyware on the phone, but this requires jailbreaking it, which puts your child at much greater risk from hackers and predators than anything they might communicate via texts. I did a quick check of Kidguard, and it has a lot of complaints. Plus the fact that kidguard gives the app provider complete access to your kid. How far are you willing to trust them?
There are a couple of roundabout ways. One is to back up the phone every day, then get a backup extractor program that can let you view the contents of the backup. Another is to get another iPhone solely for the purpose of monitoring, that you would not use for anything else. Log it in to the same Apple ID as the child's phone, then enable text message syncing. This will replicate the messages on the monitoring phone. But don't make it your phone, because your child can then spy on you. But this won't help with the other messaging apps. You can block installing apps, which will force your child to use only the built in messaging to deal with that possibility.