There is a clear difference.
If you are eavesdropping then you are listening when you shouldn't be (the conversation is private). If the parents were in a room with the door closed and the kid had his or her ear to the door to try and listen to what they were saying, then he or she would be eavesdropping.
'Overhearing' something just means that you heard it. It might be that you weren't supposed to hear the conversation but it's not your fault that you heard it. So as an example, if the parents were having an argument and the kid heard what they were saying because they were raising their voices then this is not eavesdropping, he or she simply overheard the conversation. Or as another example, when walking down the street I will overhear fragments of the conversations between other people simply because I am in close proximity to them. But I am not eavesdropping.
Now for the grammar of the sentence. You 'eavesdrop on someone.'
The child was told off (condemned is extremely strong, you wouldn't normally use it here, 'scolded' or 'told off' are more appropriate) for eavesdropping on his parents.