If you study what the Bible says about fatherhood, you realize that fathers are representatives of Christ and that Christ is actually the picture of fatherhood. The puritan author, William Gurnall (1616-1679) speaks of a father as having three roles, each of them are also the roles of Christ.
Every father hath the care of souls upon him. He is prophet, king, and priest in his own house, and from these will appear his duty.
First, he is a prophet to teach and instruct his family. Wives are bid to learn at home of their husbands (1Co 14:35); then surely they are to teach them at home. Parents are commanded to instruct their children: “Ye shall teach them when thou sittest in thine house” (Deu 11:19). “Bring them up in the nurture and admonition of the Lord” (Eph 6:4). Now, there is a teaching and admonition by prayer to God and praising of God, as well as in catechizing of them: “Teaching and admonishing one another in psalms and hymns” (Col 3:16). The father’s praying with his family will teach them how to pray when by themselves. The con- fessions he makes, petitions he puts up, and mercies he acknowledges in his fami- ly duty are an excellent means to furnish them with matter for their devotion. How comes it to pass that many...children, when they come to be themselves heads of families, are so unable to be their relations’ mouth to God in prayer, but because they have in their [childhood] lived in prayerless families and were kept in ignorance of this duty...?
Again, he is a king in his house to rule his family in the fear of God...He is to say with Joshua, “As for me and my house, we will serve the LORD” (Jos 24:15). Would it be a sin in a prince not to set up the public worship of God in his king- dom, although he served God himself in his palace? Surely, then, it is a sin in the father not to set it up in his house, though he prays himself in his closet.
Lastly, he is a priest in his own house; and where there is a priest, there must be a sacrifice. What sacrifice [is there] among Christians, but the spiritual sacrifices of prayer and thanksgiving? Thus, David went from public ordinances to perform private duty with his family: “Then David returned to bless his household” (1Ch 16:43), that is, saith one upon the place, he returned to worship God in private with them and to crave a blessing from God upon them.