It depends on the particular phone. Some phones have a built-in ability to automatically forward text messages to an e-mail, and smartphones have the ability to download apps that can do the same thing. The only catch is that most of these solutions are visible to the user of the phone, so they will know that their messages are being forwarded and be able to turn it off.
There are some third-party programs out there that give parents a great deal of control over their child's phone and aren't able to be turned off by the child. I'd suggest you look into one of them if you need that type of security.