Whose definition of it?
Heb. 1:1, "God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets."
Rom. 9:5, "Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen."
This question is the "elephant in the room" on a lot of Reformed discussions. Definitions do count and words do change their meaning.
"Gay" doesn't mean what it once did and few of us would use the word casually anymore to mean "happy," as my mother would have done in the 1940s and 1950s. That happens also in church circles. In some Roman Catholic circles outside the United States, the term "presbytery" means the house where a group of priests live, i.e., a parsonage, and distinguish it from a "monastery" because that is a place for monks who are member of a religious order and may or may not be priests, while a "presbytery" is where priests live who are, in most cases, serving as pastors of a local parish and are not members of a religious order. Catholics don't usually use the word that way in the United States because they know it will be confusing to people who identify the word with Presbyterianism, and instead use terms like "parsonage" or "rectory."
There are people in American Christianity who are using "patriarchy" as a curse word to attack anything resembling traditional conservative family life, let alone biblical standards for families. Conversely, there are people in some of the most extreme wings of conservative Christianity, some but by no means all of them Reformed, who are using "patriarchy" in ways that promote not only male headship in the home but also home churching and an elevation of the authority of fathers above church eldership.
There are plenty of people who are unaware of the way the word is being misused, or only dimly aware of it, and say things like this: "Abraham was a patriarch, so of course we believe patriarchy is biblical," without realizing there are people who say the same phrase but mean by it that Abraham, because he was the religious head of his family, the war leader of his family, and the biological father of all later Israelites, should be a model in all or most ways for modern Christian fathers. I would not be surprised if 95 percent of the people who make comments like that don't realize some of the same things are being said by people who promote "sovereign citizen" movements of people gathering guns in some remote rural homestead to fight off the "godless soldiers who may come to take away our kids," and who if they have any relationship with any church at all, are basically a group of "home churchers" who elevate fatherhood above eldership.
The people on the right who use "patriarchy" that way are a tiny minority. I know that, and most Reformed people, because of our high view of the institutional church, of office, and of ordination, ignore them. It's more of a problem in Baptist circles than Reformed circles. But as the homeschooling movement has grown in Reformed circles -- and I am absolutely **NOT** opposed to homeschooling and view it as a legitimate choice some families will make, though it was not my choice and we used a local Christian school run by an IFB church and pastor who is sympathetic to John MacArthur and views Reformed people as brothers who disagree, not as enemies -- some of the extreme stuff in the farthest right wing of the homesteading and homeschooling community has seeped into Reformed circles.
Milder forms exist. Three decades ago, before I was aware of the issues, I was contacted by a conservative Reformed church whose communion practice was to hand the elements to the father of each family and the father would decide whether his wife and children had behaved appropriately and were entitled to receive communion. I knew there was something wrong with that, but as I said, I didn't yet understand the issues and was confused by what appeared to be a conservative Reformed church that had adopted that method of fencing the table rather than having open communion or elder-supervised communion. That church had adopted their practice in an attempt to become more Reformed than their earlier practice of open communion, and I'd never seen it before so didn't understand how to respond, other than that I had a sense something was wrong. It was only much later that I realized they had adopted some (not by any means all) of the teachings of the "patriarchy" movement that has the effect of elevating fatherhood above eldership.
The proper Reformed response is to use Abraham Kuyper's distinction between the three spheres of the family, the church, and the civil magistrate. Abraham had all three roles, and Moses was both a civil and religious ruler, but ever since the replacement of the judges by the kings of Israel, there has been a distinction between the role of the civil magistrate who cannot (for example) offer sacrifices as King Saul presumed to do, and the religious leadership of God's people. Likewise, while ruling one's own household well is a requirement for the eldership, determining qualifications for communion belongs to the eldership, not to individual fathers, and there are good reasons why a father who is an elder SHOULD defer to his brother elders if there are questions about whether his son or daughter is ready to make profession of faith. Contra the teaching of some in the "Patriarchy" movement, Abraham is not our model. The model for the organization of the New Testament church is found in I Timothy and Titus, as well as other related passages which (among other things) show the need for a plurality of elders in the local church and that the episkopos (bishop) and presbuteros (elder) are the same office, so the pastor should be first among equals on a session, consistory, or board of elders. Elders run the church, and while fathers head their home, they are accountable to the church elders on church matters.
Where things get messy is that a tiny minority that most Reformed people consider irrelevant, and mostly exists outside Reformed circles, is being used as a cudgel by true liberals and radical feminists to attack "normal" views in conservative Reformed circles of male headship in the home. I guess the idea is that if someone teaches that men are supposed to head their homes, they're secretly gathering guns in their basement to plot a violent revolution.
It sounds crazy, until we realize that "crazy" really does exist.
I live in the Ozarks. I've seen the damage caused by extreme teaching in house churches and a view of Christian education that goes way beyond anything Doug Wilson teaches. Put bluntly, local law enforcement tell me there are places where they will not go without full body armor and rifles not just in the trunk of their patrol car but ready for use on their front seat, and with a SWAT team ready for backup. I'm no fan of the abuse being done by "Child Protective Services" in some liberal states, but around here, likely examples of abusive homes are being avoided because most of our law enforcement and government officials are conservatives who support family choice and don't want to set off a "Ruby Ridge" scenario with false accusations against a homeschooling and homechurching father in the patriarchy movement unless they have absolutely crystal clear evidence of sexual or physical violence by a father against his family in their off-the-grid homesteading compound out in the woods.
Here in the Ozarks, and in a lot of rural communities, there's a long history of "live and let live" with various right-wing fundamentalist groups being told, quite correctly so, that the civil government won't bother them and they can do what they want. Almost nobody today wants the nightmares that happened when I was a young man and saw homeschoolers fearing truancy officers would raid their homes and take away their kids.
But as Reformed people, we can't ignore the fact that there are people on the fringes of Reformed Christianity, most but not all of them outside what most of us would consider to be confessionally Reformed church life, who are using "Patriarchy" as a model for something which Reformed churches haven't usually had to deal with.
Just as I was confused three decades ago by a Reformed church that was handing decisions on communion qualifications to the fathers of the congregation, there are Reformed people in traditional Reformed circles who are facing things that most of us have never had to deal with before, but which have been going on for a long time in American fundamentalist circles.
Trust me on this -- I am not an enemy of fundamentalists any more than Machen was. I want to see fundamental Baptists become Reformed. (And yes, I'm happy if they become Reformed Baptists -- I support infant baptism but view Mohler, Spurgeon, Bunyan, etc., as brothers in the Reformed faith.)
But as we see more and more people brought into Reformed circles by the "New Calvinist" and "YRR" movement, we're going to have to deal with people who have very different views of the relationship between family, church and state than most Reformed people have had.
We need to know how to respond to those for whom "Patriarchy" means something quite different from how traditional Reformed people may use that word.