I suspect that if most homeschools are like mine, the mother in the household is responsible for most of the actual teaching. There are exceptions of course, but I would say that this is true most of the time, especially because homeschooling families with stay-at-home moms often rely primarily on one income. So, it would be easy to suppose that fathers are by-and-large irrelevant when it comes to the day-in and day-out success of academic work. -- Wrong conclusion. Without the father's role being filled (either by the dad himself, or a godly substitute* in the case of widowhood or divorce), the result looks more like a tug of war than education.
To make an architectural analogy: In the structure of the homeschool, the Word of God is the foundation, the father assumes the role of the roof and walls, and the mother handles most of the interior decoration and arrangement. Without the covering of the father/husband there isn't much to decorate or arrange. The storms of life (nicely provided by the world, the flesh, and the devil, not to mention those problems sent by God to further our sanctification), are weathered by a strong outer structure so that those in the interior stay dry and safe.
To be sure, when it comes to the academic or business success of my older children, I am often credited with having given them a good foundation. And this is true. But, what remains largely unheralded and overlooked is that without my husband supporting and encouraging the entire enterprise, the fruit wouldn't be as good, and I would most likely be called a "former homeschooling mom" rather than the active one I am today.
All this being said, it is vital for the husband/father in the homeschool to be an expert when it comes to God's law-word and its application to his family. Over the years, many a dispute that has arisen between my children/students and me has been astutely handled by my husband's patient listening, followed by insightful comments that more often than not, open the door to resolution. These were not whimsical solutions, but ones where God's ensciptured Word was given prominence and authority.
Lastly, I would be remiss in failing to acknowledge a bit of envy that often permeates my soul when it comes to my husband dealing with our children. From the time they were very little, the deep sound of Daddy's voice was something that produced immediate change that mine never seemed to accomplish. I've always wanted that deep, male-voice when it came to getting results or altered behavior in the kids. And this phenomenon didn't change when they got older. Even as some were feeling their oats, nothing brought them back down to earth faster than the God-given authority of their earthly father using his "get your attention" voice.
So, here's a round of applause for all the thousands of homeschooling dads who propel one of the most significant movements in our time -- one that is sure to realize the furtherance of the Kingdom of God on earth!