Summary: Some authority is built into the fabric of God's very creation, and our response to earthly authority - whether we wield it or submit to it - reflects our acceptance of God's authority.
Many scholars and preachers treat today's passage as an extension of last week's exhortation for husbands and wives to submit to one another. They call the section between chapter 5 verse 21 and chapter 6 verse 9 "house codes." Paul almost certainly drew from well-known early Christian teachings in drafting these verses. They are important largely because Christians - like Jews - were seen as subversive elements in society and especially as a threat to the family structure. Christians needed to show they did not threaten order and decency. And to some extent these passages were written to counter slander and accusations. And of course when Paul first wrote this, he didn't put so much as a paragraph break between the first verse in this chapter and the last verse in the previous one, let alone start a whole new chapter!
But I think that the break is important, because there is a fundamental difference between the instructions on how husbands and wives should relate to each other, and on how parents should treat their children, and how to manage employee-employer relations. Because as we saw last week, the relationship between husband and wife is not ultimately one of subordination but of equality. Although in both law and in custom men had authority over women, God's purpose in creation is for equal partnership, equal dignity, equal submission. But children and laborers are not in the same position. In fact, culturally their status was even lower than women. But instead of turning the relationships upside down, Paul affirms the authority inherent in these relationships.
Now, Paul doesn't tell the Ephesians - and by extension the other churches in the Mediterranean world - to stop freeing their slaves, or to stop treating their slaves like brothers and sisters in Christ. Nor does he go back on other statements he makes elsewhere making slave and free equal in the sight of God. Instead, he gives both fathers and children, slaves and masters, theological motivations for not shaking up the social applecart too vigorously. It is not just out of a desire not to scandalize the neighbors, but a recognition that in these relations - even when redefined through a relationship with Christ - there is lawful authority.
Children are told to obey their parents. Paul doesn't say, "obey your parents if they are Christians." He says, "Children, obey your parents in the Lord, for this is right." [v.1] Obedience to parents is part of obedience to God. God built parental authority into the created order as an essential ingredient of social stability. But just as husbandly authority has been abused since the fall, so has parental authority. Parents could kill their children, or sell them into slavery or prostitution. Exposing unwanted children - that is, leaving the newborn out in the open either to die or be picked up by a soft-hearted stranger - was common practice. Unfortunately, part of our society's reaction to overly harsh treatment of our children in the past has led in many cases to parental abdication of their god-given responsibilities given originally in Deuteronomy,