Yes! You know, when I was a child my parents knew what I was reading. They took me to the library. They often read along on the books that I chose. Doing so, they encouraged me to read, while ensuring they had at least some influence on what I might take away from what I was reading.
In my teen years, that sort of direct involvement lessened, but it never went away altogether. Even though I disagreed with a lot of my parents beliefs, I never bothered trying to hide books from them. That never occurred to me, because as conservative as my parents were, they believed reading was critical to growth and almost sacred in terms of their democratic values.
But back to my earlier childhood when my parents did actively supervise my reading, why don't conservative parents just do that today? And yes, mobile devices are a big part of that.
If parents believe that they can help their children grow by not allowing them to read books they disagree with, I wonder how they're handling mobile devices.
I don't know, though I know progressive parents who keep a pretty tight reign on how their younger children consume media on the Internet.