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.

He overheard their conversation. This means that he just happened to hear what they were saying, because he was sitting close to them or they were speaking loudly. He wasn’t making any conscious effort to hear what was being said. He was eavesdropping on their conversation. He was deliberately trying to listen in on their conversation, perhaps even moving closer to them so that he could hear better.

To eavesdrop is to listen secretly to a private conversation, without their knowledge. To overhear is to listen to something without intending to, without their knowledge. Examples They were eavesdropping on our conversation I overheard a funny conversation on the train today