I'm not aware that the Bible tells us this. Parents are a team working together to raise the children (Eph. 6.1-2). If parents are to disagree about this, just like anything else, they need to work it out.
Let's look at a few texts.
Ephesians 4.21 says that husband and wife submit mutually to each other. No person is made ruler over the other. In the following verses, the wife is to be subject to her husband in everything, and he is to completely and sacrificially give himself up for her, which is also a form of submission in everything. He is to die for her if needed.
Ephesians 4.23 says the wife is the body and the husband is the head. The point is not that he is the boss, but rather that together they form one flesh (v. 31).
Ephesians 4.25, 33 commands the wife to respect her husband. It's not obedience, it's respect. And the husband is commanded to love his wife, which is a posture of respect for her.
Gene Veith writes,
"There are roles within family. There is authority in family. But at the same time, Christian books tend to reduce things to, “Who has to obey whom?” it reduces roles to power relations, whereas the Scriptures and the doctrine of vocation teach that the purpose of every vocation is to love and serve your neighbor. When we forget the mystery of how God works in vocation—that it’s about loving and serving—we end up with a legalistic set of rules. That’s what happens when the gospel is drained out of our view of vocation."
The purpose of the Ephesians text is not the put the woman "in her place," but rather to put Jesus in His place.
Kenneth Wuest writes,
"If the husband’s supremacy had been in view, it would have been expressed by tois kuriois (your lords and masters)."
Liefeld writes,
"There is no word of male rulership in the text. Context is of the filling of the Holy Spirit (v. 18) and mutual submission."
In Ephesians 6.4, fathers are commanded not to exasperate their children. This is not a suggestion that the father only is regarded as the boss of the children, as vv. 1-2 put both parents as equals over the kids. The father is portrayed, not as an absolute ruler, despot, and dictator, but as one who considers the wellbeing of the child, and in that sense lives to serve them as Christ served the Church to maker her holy.