What is the ruling regarding a parent who asked their child to swear on the Qur’an, then discovered their lie, and now feels regret and anxiety?

The action mentioned is a sin, and the child is not required to confess it. Rather, he should conceal himself and the girl, based on the hadith: "Whoever commits any of these abominations, let him conceal himself with the concealment of Allah." If he swore falsely, he must repent for the act and for the false oath. The real solution is to marry him off and help him remain chaste, to occupy his time with beneficial activities, and to admonish him to soften his heart.