The family of the just man who lives by faith

But in the family of the just man who lives by faith and is as yet a pilgrim journeying on to the celestial city, even those who rule serve those whom they seem to command; for they rule not from a love of power, but from a sense of the duty they owe to others - not because they are proud of authority, but because they love mercy (Augustine, City of God p693)

To live by faith as a father is to "serve those whom they seem to command." I would interpret Augustine's "duty they owe to others" to be the debt of love, not an obligation of service. The great responsibility of a father is to live as a pilgrim, serving his family, and doing all within his power to see that his household love God.