I am sure you already know this. Well I guess I have put in my two cents for the day. It doesn't matter what century we live in. The Bible is our guide for our lives. It doesn't change from century to century, even if our cultures do.
Christians are to obey the Bible.
The Bible tells us also to obey the authority over us. But clearly, if there is a government law that directly opposes God's Word, then we must obey God and not man. The Bible, especially in Proverbs, clearly spells out how to train up our children. The lack of the rod results in disobedient and unruly children. Using the rod in anger results in rebellious children.