The report, produced by the charity Working Families, the Global Institute for Women’s Leadership at King’s College London and the University of East Anglia, said there were "huge barriers" to flexible working.
Under UK Government proposals, British workers will be able to request the right to work flexibly from their first day in a job, instead of after six months, which is the existing rule.
Previous research by the TUC has found one in three flexible working requests in the UK are turned down.
The report says that normalising flexible working would help working parents in particular, as many – especially mothers – have to sacrifice pay and progression to secure flexibility, leaving them stuck in jobs that may be flexible, but have few opportunities for advancement.
This stems from employers’ negative attitudes towards part-time and flexible working arrangements, and a perception that senior roles cannot be performed flexibly, the researchers say.
Gendered assumptions by employers – such as that mothers are happy to forgo training and progression, and that fathers do not have family responsibilities – also play a part, while flexibility is often defined only in narrow terms, failing to offer the reduced hours and alternative schedules that some parents need.
The report, which held focus groups with working parents, emphasises that job and financial security, control over when and how work is carried out, and support from managers are all valued by parents and must be combined with flexible approaches.
Jane van Zyl, chief executive of Working Families, said: “Over recent decades we’ve seen plenty of progress in parents being able to access high-quality work, but we know that some barriers to change have remained stubbornly in place.
"The juggling act parents carry out on a daily basis is much discussed. This new research brings it to life by looking at the quality of the jobs parents can get, and the compromises they make to try and secure them alongside meeting their caring responsibilities.
“Working parents are often undertaking low-paid roles, well below their level of skill, and giving up on opportunities to progress or take promotion because of their childcare responsibilities.
"It shows very clearly that work and care are not easily reconciled without flexibility, security and supportive employers.”
Dr Rose Cook, senior research fellow at the Global Institute for Women’s Leadership, King’s College London, said: “Our research highlights that improving parents’ working lives and keeping them in employment relies on far more than just allowing them to request flexible working from day one.
"Flexibility as it currently stands arguably bolsters gender inequalities rather than tackling them, since it is so often associated with poorly valued, marginalised roles, and is unavailable within higher quality jobs.
“Mothers especially should not be having to sacrifice fair, fulfilling and meaningful working lives so that they can manage caring responsibilities. Both structural and cultural changes are needed across the workforce so that these trade-offs don’t need to be made.”