Well, I can't quite figure out:
Proverbs 26:3 "A whip for the horse, a bridle for the donkey, and a rod for the fool's back."
Proverbs 19:29 "Judgments are prepared for scoffers, and beatings for the backs of fools."
Proverbs 26:3 is dealing with control or the giving of direction. A rider of a horse can give directions to his mount with a small horsewhip. Even though the horse is much bigger than the man and the strikes of the whip are not much more than an annoyance, yet the horse will comply with the rider's will. Donkeys are notoriously stubborn beasts, yet a rider can direct a donkey with bridle about the donkey's nose. Again, the donkey is stronger than a man and more stubborn than most men, yet a small tug with a bridle will win out in the end. As with the first two, so is the last. A spanking with a switch (rod) on fool's backside will cause him to give up his foolishness and be obedient. The pain of being switched seems a minor annoyance, yet it can cause a change in behavior. Just as a horse whip is well-suited for directing a horse and a bridle is well-suited for directing a donkey, a switch is well-suited for correcting a fool. There is also a more subtle statement being made as well, fools are little more than brute beasts in their behavior.
Proverbs 19:29 is dealing with the certainty of consequences. Scoffers are fools who scorn authority, but their disdain for authority will not prevent them from facing a judge when they break the law. In a sense, the law with its recommended sentences (judgments) was written for people who believe they can ignore the law (the scoffers). In the same manner, fools act without thinking about the consequences of their actions. Just because they believed there wouldn't be a consequence, it doesn't mean they won't receive one. The use of spanking the backside of a fool is laid out in advance because fools exist, doing things without thinking. This verse is comparable to "For rulers are not a terror to good works, but to evil. Do you want to be unafraid of the authority? Do what is good, and you will have praise from the same. For he is God's minister to you for good. But if you do evil, be afraid; for he does not bear the sword in vain; for he is God's minister, an avenger to execute wrath on him who practices evil" (Romans 13:3-4).
Finally, in both of these verses, there is no mention of the parent-child relationship. The implication is that switching could be a legitimate form of punishment in other cases where an authority, such as a government, is in need of disciplining a foolish member of society. This is not a general authorization for anyone to take such authority upon themselves, but it does mean those countries that do use switchings as punishment for certain crimes are not as barbaric as some claim.