The correct option is
A have eyes in the back of their heads
'To have eyes in the back of their heads' means 'to know everything that is happening around them'. When used in the given blank, the conversation means: 'There are new developments in the cyber world every day, some of which, could be unsafe for children. So, parents have to constantly update themselves about all the developments in order to know if their children are safe'.
This expression meaningfully completes the given sentence. Hence, option A is correct.
'To throw caution to the wind' means 'to act in a completely reckless manner'. In other words, it means, 'to do something without worrying about the risk or negative results'.
'To skate on thin ice' means 'to be in a precarious or risky situation'. Thin ice is extremely dangerous to skate on as the ice could break and one could fall into the cold water below. Thus, this expression derives its meaning.
'To save one's own skin' is' means to rescue or protect oneself from danger, trouble, or difficulty, usually without concern for the welfare of others'.
Given the meanings as above, these expressions cannot be used to complete the given conversation meaningfully. Hence, options B, C and D are incorrect.