Compare some scriptures about fathers and children: What father gives his son a stone? (Matt 7:9); and how are we to care for our children? (Eph 6:4); (Col 3:21). And an example of it: 1 Thess 2:11.
Further on the Proverbs passage from Thomas Haller:
Through a closer examination of the Hebrew word for “rod” (shebet) one can see that in the Hebrew dictionary it has various meanings, a stick for walking, writing, fighting, ruling, and punishment. The word “shebet” is most frequently used when referring to shepherds who are tending their flocks. The Shepherds used the stick to fight off prey and to gently guide wandering sheep, not to beat them. Please remember that these verses come from a book of poetry. Writers of poetry use familiar words of the day to represent concepts that the people to whom they are writing can create an image of what they are writing about. The image that we are to extract from these verses in Proverbs is one of creating a culture of accountability. The point that God is making is that we as parents are to hold our children accountable for their choices and actions.