It is not necessarily true that internet security should be taught in this way. We can certainly argue that internet security should be taught by parents and not by teachers. Schools should be places for learning academic subjects, not for learning things like how to protect yourself from internet malware.
However, if we assume that this subject should be taught as a partnership, it is because the subject needs the input of both parents and teachers. Not all parents necessarily have the technological savvy to teach their children to be safe on the internet. Even if parents lack this savvy, their children still need to know about internet security. Therefore, it is necessary to teach this at school. However, it is also necessary to teach it at home. Much of internet use (particularly that use which does not go through school webfilters) occurs at home. Different parents have different ideas as to what their children should be doing on the internet. Therefore, internet security must be at least partly taught at home.
For these reasons, we can argue that both teachers and parents must be involved in teaching internet security.