Parents influence children? -- Terrorism angle

There have been quite a few reports on young people from Australia, Indonesia, Malaysia and the UK going to Syria to join ISIS after being influenced by social media, to the extent that their parents were shocked when it was uncovered that their children actually have such terrorist ambitions.
We can use this to show the limited influence of parents on children, as nowadays, young people tend to get their influence from social media and celebrities, religion and even strangers -- provided these strangers can gain their trust on social media. The argument that parents have great influence on their children is based on the assumption that there are close ties btw them and children look up to their parents, but this is no longer a given in today's society, esp when applied to older children.
Note that the names of the specific young people are not impt here as they are not that famous, but that collective identity as youth who go behind their parents' back to join terrorism is what interests us here....
But of course, on the flip side, there are also cases of parents who brought their children to Syria to join the terrorists. see HERE, and HERE.  

Qn: "Parents have (too) much influence on the lives of their children." Do you agree?