Dads who want to be more involved in the care of their children fear that asking for more flexible hours might damage their careers, the chairwoman of a new probe into the issue says.
Conservative MP Maria Miller says such requests can even lead to employers questioning their workers' commitment.
Research suggests 44% of dads have lied about family-related responsibilities.
The Women and Equalities Committee inquiry aims to find out how much support fathers receive at work.
On Wednesday, the committee began taking evidence to try to uncover what demand there is for change.
MPs will look at how well fathers feel their current working arrangements help them to fulfil caring responsibilities for children of all ages - and if they have the financial support to carry these out.
The inquiry comes in the wake of the 2017 Modern Families Index, authored by employment campaign group Working Families, which suggested that while family was the highest priority for fathers, half of those interviewed felt their work-life balance was increasingly a source of stress.
It will also consider:
- whether an increase in freelance, agency or casual working might have an impact on fathers
- whether workplace attitudes about fatherhood need to be challenged
- if so, what role government, employers and other stakeholders can play in overcoming them
Sarah Jackson, chief executive officer of Working Families, gave evidence on how current laws on paternity leave could be improved.
She told the select committee that paternity leave and shared parental leave should be a "day one" right for all fathers.
"At the moment you have to be an employee, and you also, even as an employee, have to have been employed for 26 weeks - at last 15 weeks before your baby is due - to be able to give notice to take two weeks' paternity leave."
"So there is this huge inequality that fathers face in the workplace, right at the start."
She also argued that flexible working laws needed to be looked at.
"Hardly any flexible working cases ever get to tribunal. There are two reasons for that. One, is that the tribunals can only look at the process, has the employer followed the proper process; they can't look at whether there is really a business case for refusal.
"The other thing is that the remedies are so low. The maximum remedy is 8 weeks' pay, so if you are a low-paid father, you are not going to gamble your job on 8 weeks' pay, and if you are an employer who doesn't really want to have to think about this, 8 weeks' pay is not much of a penalty to worry about."
However, she did not feel that making paternity a protected characteristic, which would make it illegal for companies to discriminate against men for being fathers, would work.
She said: "What I would be much more interested in talking about is some sort of protected characteristic that is about having care responsibilities.
"Because that would capture men, women, same sex couples, adopters, people caring for disabled children, people caring for elderly parents, people caring for terminally ill spouses. There are so many people who are not covered by discrimination law as it currently exists, and we could do so much better."
'Culture change needed'
Tina Miller, Professor of Sociology at Oxford Brookes University, noted that while the introduction of Shared Parental Leave was an important signal of a commitment to change, it was simply not the case that men and women now had an equal choice over who would work in a family.
"It's much more complex than that, because people need money when a baby is born, usually because of the motherhood penalty, when the mother doesn't earn so much, there are all sorts of financial reasons it makes sense for the father to stay in work."
The government forecasts that between 2% and 8% of eligible fathers will take up Shared Parental Leave, a flagship policy introduced in 2015.
Maria Miller said: "Many fathers want to be just as involved in their children's lives as mothers do, which is good for children too."
But she said there were "significant questions about whether culture at work has changed enough" to enable the shared leave policy to be effective.
And she added: "We need to find out what we can learn from other countries."
Sarah Jackson and Professor Tina Miller were joined in giving evidence by the Centre for Social Justice's Edward Davies and Family Initiative's Duncan Fisher.
Working fathers will also be detailing their experiences during sessions after Easter.