"authority" refers to those who you should obey because they are seen to have knowledge and capability which makes their decisions more important than your own thoughts or decisions on the same thing. Well, at some point, we all figure out that most authority is not deserved, and those we have granted authority are not even close to perfect and are thus often wrong, and perhaps I myself am in a better situation to decide, instead of them.
The sentence means that you have always believed that parents are always right (because that is how human children develop), but are finally starting to realize that they are NOT always right. If they are not always right, how can I be sure that they are right about this specific thing, or not? You question whether what they say is actually right or not.