Earthly fatherhood is a reflection of God the Father.  Each person's first impressions of the fatherhood of God comes from his or her earthly father.  So the first three commandments deal with our relationship with God, and the first after that deals with our relationships with our earthly parents.

How we treat our parents is related to how we treat God.  We can't control how our parents treat us, but we can control how we treat our parents.  The purpose seems to be to form us in a humble obedience before God, which should carry over into our respect for earthly authority, especially that of our parents.