A former boss once put a proposal to us. He was happy to introduce a paternity pay deal equivalent to the deal given to mothers. However, it would mean maternity pay was reduced from six months at full pay to a system of three months for both mums and dads. It was up to us. And the staff – with male colleagues fully aware of the wrath they might face from female colleagues – roundly rejected it.

Ten years later, reading a new report on the discrimination family-conscious fathers face in the workplace, I’m reminded of this episode. I suspect my boss – who had himself taken time out to care for his kids so his partner could focus on her career – knew the indignation his proposal would meet, given it would improve the lot of fathers on the payroll, at the expense of making mothers with partners at other workplaces worse off. But as a thought experiment, it served its purpose. It made me ask: are there benefits – perhaps the notion we have the automatic right to primary carer status – that women might have to sacrifice in the struggle for equality?

The report by the Commons equalities committee outlines how hard it can be for fathers in male-dominated – read macho – working environments to ask for parental leave. Dads who worked part-time to accommodate childcare told the committee of finding themselves mocked by co-workers.

It might be hard to get our heads round the idea that daddy discrimination is an important frontier in the fight for gender equality. It’s predominantly mothers whose earnings potential is hit by having children; whose career progression is choked off as a result of going part-time. In contrast, men actually get a fatherhood pay bonus. So why on earth are feminists at the Fawcett Society arguing that more cash should be channelled to dads through better paternity pay?

That fatherhood bonus is the flipside of the entrenched gender stereotypes that fuel the pay gap. Dads get a pay boost because their employers expect them to conform to the male breadwinner model, putting in longer hours to advance their pay, while their partner looks after the kids. Like mothers, fathers trying to do the right thing face discrimination.

We should learn from Iceland: three months of leave should be reserved for fathers

Some argue that the fact that women shoulder the burden of caring work is simply a reflection of individual choices. Please. Some women may indeed embrace giving up their career for family, and good for them. But plenty more are forced to by a system that shores up the male breadwinner model of family life. Shared parental leave, which allows mothers to transfer their maternity leave to fathers, was only a baby step forward. There’s a big catch: paternity pay is much lower than maternity pay, and men earn more on average than their female partners. This double whammy means families take more of a financial hit if fathers take time out. Take-up of shared parental leave has, unsurprisingly, been very low. And why is there the presumption that shared parental leave sits with the mother to gift to the father?

So we remain stuck in a vicious cycle that is reinforced by government policy. Because dads earn more, they are less likely to take time to care, which means dads continue to earn more. But research highlights the importance of breaking the cycle: when men take leave early in their child’s life, it not only results in better developmental outcomes for children, but means they are more likely to do their fair share of domestic work later.

We won’t eliminate the gender pay gap without making it more culturally acceptable for fathers to do more. Fortunately, there is a tried and tested way of achieving this: to provide a significant chunk of “use it or lose it” leave reserved for new fathers. Several countries have introduced this to great effect; in Iceland, mothers and fathers get three months of leave each, paid at 80% of average earnings, with a further three months to be allocated between them. The vast majority of Icelandic fathers take their reserved leave. It is little wonder that Iceland has topped the global gender gap rankings for the last nine years.

We should learn from Iceland: three months of leave should be reserved for fathers. If it was paid on the same basis as the first three months of statutory maternity pay, but capped at median male earnings, it would cost between £200m and £400m a year, far less than the government’s marriage tax break, and a far more effective way of promoting healthy family relationships. Much of that cost would likely be recouped through higher productivity and increased female earnings.

This might mean employers insist that statutory maternity leave in two-parent families shrinks from a year to nine months. But that’s a price we have to pay for expanding the choice of women who don’t necessarily want, but feel forced, to be the default carer. And anyway, social attitudes are changing: just under half of millennial fathers say they would take a pay cut to achieve a better work-life balance, and more than half of men say childcare should be shared equally.

Women’s secret weapon in the fight for equality could be arguing for better workplace benefits for fathers. We have to make it OK for men to embrace an approach to work-life balance that’s closer to that taken by women. Otherwise gender equality will remain a distant dream.

• Sonia Sodha is an Observer columnist