My gut reaction is that parents looking through their children's phones, apps, messages and photos is a huge invasion of privacy. As the article suggests though, there's also the need to protect and manage what our children are involved in, just like offline activities.
There's something about smartphones that is so personal, that there doesn't quite seem to be a good comparison for pre-technology privacy. I don't think I'd have a problem with dictating what my child watches on television, eats, drinks, or otherwise consumes - but controlling smartphones seems something more akin to thoughtcrime.
I was lucky enough to grow up when technology and connectivity was so new that apparently it wasn't used in such a mainstream sinister way as it appears to be now. Going through puberty seems like an even more harrowing experience (ala the move Eighth Grade) with the internet, that seems to go quite unnoticed.