The healthcare sector is least likely to offer flexible hours, leaving parents "resentful" a report has found.
Parents of children under the age of 16 or disabled children under the age of 18 are currently allowed to request flexible hours.
The new Children and Families Bill will extend the 'right to request' to all employees.
The report, released by the Working Families organisation, showed that education, retail and healthcare were least likely to offer staff flexible working.
The majority of responsibility for emergency childcare still falls on the mother, but young fathers aged 26-35 are almost as likely to be called as their partners.
Young fathers were particularly resentful about the lack of choices over remote working and felt this was having a negative impact on their career progression.
In addition, fathers with a single child tended to be more resentful towards their employers than those with more than one child.
Sarah Jackson, chief executive of Working Families, said: “It is clear that expectations for work life balance are changing, particularly among younger working fathers. This is a quiet revolution in attitudes, which may have long-lasting impact in the workplace.
"The male employee, focused full-time on his work, is becoming a museum piece. Tomorrow’s workers, male and female, will expect time and space for their family lives and responsibilities alongside their work.”
Denise Keating, chief executive of the Employers Network for Equality and Inclusion, said: “It’s a shame that the legislation that will allow the right to request flexible working for all could be delayed. Moving towards flexible working for everyone is important in this ever-changing world. Leading employers recognised this many years ago and are already offering formal and informal flexibility to their staff."