I guess I came in late for this controversy: from 2008 there’s a YouTube clip of a famous preacher and his wife, responding to the question: “What are your thoughts on stay-at-home dads if the woman really wants to work?”
“Too many guys take too little responsibility” was part of the answer, one with which I fully resonate. We have a culture where men play at being boys well into their adult years. At a time when their fathers and grandfathers had buckled down to marriage and a job or were off fighting Nazis, some guys focus on playing the field or playing paintball until they’re, well, practically my age. The women are complaining, as they ought to be. These guys need to hear a Word about their behavior.
But let’s put them to one side, since the gist of their response was something else: If men are not the primary bread-winners in the family, they are not doing “what the Word says.” Parenting must be done principally by the mother, not just “anyone,” not even the father. The idea of a father staying at home to focus on raising children is a perverted idea, taken from our modern culture, not the Bible. These men are “conformed to this world.” Such behavior would even by “a case for church discipline.”
Okay, let’s see what the Word really says. They based their opinion on 1 Tim 5:8 – “But if anyone does not provide for his relatives, and especially for members of his household, he has denied the faith and is worse than an unbeliever” (1 Tim 5:8). Here the ESV version uses masculine pronouns (his, he) to represent what is really generic language in the original. Our case is a prime example of how we read verses as applicable to men, when in fact they apply to everyone. Here “his or her” would be an appropriate translation: men and women are worse than unbelievers if they do not provide for their household. 
Let’s go further: in 1 Timothy, as in all the New Testament, “household” does not mean the nuclear family of dad, mom and minor children. Households in that time typically included the extended family of in-laws, uncles and aunts, grandparents and cousins. In fact, this passage demands to be read that way: it refers to a widow who is asking the church for financial support, but her Christian relatives – perhaps people who live under the same roof! – aren’t taking care of her. Those children or grandchildren should “care for, or provide for” their mother or grandmother. Even your pagan neighbors might be offering their aged parents support and a roof over their head – will you do less than they?
1 Tim 5:8 speaks to men and women who think the church should give their aged relative financial help, while they themselves have not done what they can. It has nothing to do with who is the breadwinner. It has nothing to do with male and female family roles. It doesn’t even have to do with earning a salary and sharing it.
The picture we have of Dad going off to work while Mom stays at home with the children, even teenaged kids, is hardly a biblical one. It owes as much to culture since the Industrial Revolution as to the Scriptures. In 1st-century society there would be less division between bread-winner and stay-at-home parent. Fathers usually worked in or near their homes and spent much of the day with their children nearby. When the boys were young teens, they would work alongside of their fathers rather than spend time at home with their mothers – past a certain age, fathers did the primary raising of the boys, mothers of the girls.
When Paul said that women should be “busy at home” (Tit 2:5), he didn’t mean as opposed to working at a job, but busy as opposed to gadding about, not in their own house, but wandering “from house to house” (see 1 Tim 5:13; might a modern parallel be, “wandering around other office cubicles rather than tending to her own”?). Even within Paul’s circle of friends, women worked as the primary bread-winner (Lydia, Acts 16:14, whose trade in super-luxurious purple cloth probably means she was an internationally networked businessperson) or worked side-by-side with their husbands (Priscilla, Acts 18:3, who helped make tents – decidedly a step below Lydia on the economic ladder!). If Priscilla and Aquila had children – and there’s no reason to supposed they didn’t, the New Testament usually doesn’t mention if a family has kids – then both parents were around to raise them. I see no verse where Paul takes Aquila aside to rebuke him for raising children or permitting his wife to work.
There are too many cases of men who are lazy or lack direction; we are up to our chins with Peter Pans. But let’s not hassle those real men among us who choose to raise their own kids.
 The other half of the question was “Or even if both want/need to work?”, but this part was not addressed in their answer. [Update: I guess it’s okay to reveal that the preacher in question has since lost his ministry, in part because of what some (myself included) perceived to be his anti-women bias. However, my observations here have to do with this particular approach to parenthood, not with any one preacher’s articulation of it.]
 For readers of Greek: 1 Tim 5:8 had τις/tis, the indefinite pronoun being correctly rendered as “anyone” in the ESV, so as to refer to men or women. Nevertheless, the ESV goes off-track with των ιδιων/ton idion, which does not mean “his” household, but is generic, “his or her household.” The main verb ου προνοει/ou pronoei should not be rendered “he has denied the faith;” this time, the verb is generic (no subject is stated), meaning “he or she has denied the faith.” Paul is thinking back to 5:4, correctly rendered in the ESV as “if a widow has children or grandchildren [in the Greek, the “children or grandchildren” could be male or female], let them first learn to show godliness to their own household and to make some return to their parents.” Thus: men and women should provide for their families. The NRSV of 5:8 has “And whoever does not provide for relatives, and especially for [immediate] family members, has denied the faith and is worse than an unbeliever.” This is not a “politically-correct” or unisex translation of the verse, but a solidly literal one.
“Can stay-at-home Dads be “real men’?” by Gary Shogren, PhD in New Testament Exegesis, Professor of New Testament at Seminario ESEPA, San José, Costa Rica