What a foolish person you are, thinking that every child will respond exactly the same way to a parent's disciplinary action.
Some kids will tell the truth when asked. Some will tell the truth if pressed a bit. Some will tell the truth when they feel they can't pull off a lie. Some will tell the truth when it is overwhelmingly apparent they can't pull off a lie. Some will not tell the truth until they are caught after the fact. And then, some do not even do that, and proceed into a whole other new lie to try to explain the first lie.
And that is the case of this girl here. In fact, the article doesn't even state that she did tell her folks she was lying to them, it is just known that she did and they knew she was lying.
Why would it have been so much better for the girl and her parents if she just continued to lie to her parents? What is the great lesson she is learning, other than that if you can get away with lying to legitimate authority figures, it's okay?
Yes, my wife and I are foolish people in expecting to raise honest children who are now in their late twenties, both college graduates, never ended up in jail and have excellent careers.