Other times they say, "TV is dangerous. My child doesn't know enough not to listen to it."
These two statements represent opposite views about ignorance. One view is that ignorance cause stubbornness and a closed mind. The other is that it causes gullibleness and an open mind.
So, which is it?
Suppose it's the closed mind. Children are born with a very closed mind. Whatever you do, and whatever they are exposed to, it won't make much difference. Slowly, they will become slightly more open minded, and learn a little. The older they get, the more easily they will learn new things. By the time they are 40 or 50 years old, they will finally be very open minded and learn new things all the time. I think we can see this is not what the world is like. People learn more rapidly when they are young. They appear to get more closed minded with age, not less.
Now onto the possibility that children are born with a fairly open mind. Then, when a child doesn't listen, one has to wonder why. (Asking why a child does or thinks something is a major theme of TCS.) He wasn't born closed to what his parent is saying. If he's rejecting it, there is some cause after his birth. It could be a history of his parent's advice being unpleasant for him, or it could be that he has a different (contradictory) idea he thinks is superior, or it could be that he's trying to listen but there is a failure of communication (e.g. the parent's explanations are too complicated and confusing, or too dumbed down without enough persuasive content).
If the child doesn't listen because listening has gone badly for him in the past, that is a problem the parent can and should do something about. He needs to take all of his advice and consider it carefully from the point of view of whether it will be pleasant for his child.
If the child doesn't listen because he has a different idea, the parent can talk to him about what his idea is, and offer criticism of it, and ask questions about it (the child could learn a lot trying to answer questions about it). The parent could can also accept criticism of his own idea from the child. That way the child will learn to think of criticisms, and see which ones work and how well (some criticisms will result in a short explanation of how they are trivially mistaken, some will lead into a whole new area of interesting discussion, and some will lead to the parent changing his mind).
And if you're going to have a discussion with questions, criticism, new ideas, and explanations being exchanged, then that is just as if you were having a genuinely open-ended discussion where the final answer isn't a foregone conclusion. So as one final step, the parent should himself have an open minded, and it should really be a truth seeking discussion, instead of a "how to make the child listen to the idea I already KNOW is right" discussion. If your idea is right, it will be the conclusion of a truth seeking discussion anyway, so you don't have anything to lose.