Two thirds said it would be helpful to work flexibly as their children became older, and half of parents whose children were aged between 11 and 16 said they would help with homework if they could find the time.
Ms Harman said: "Children don't stop needing their parents' time when they reach their sixth birthday.
"We have already built a strong foundation of support for families through the right for parents with children under six to request flexible work.
"But, as any parent knows, older children going through the teenage years need just as much support and guidance."
She later told the BBC: "It means employers can't say 'I can't be bothered to think about that, these are your hours, you can take it or leave it'.
"It is reasonable for employers to recognise that a lot of people they're employing are parents as well.
"They're fathers of children, they're mothers of children, and it's in everybody's interest that the next generation are able to be properly brought up by their parents."
Sarah Jackson, chief executive of campaign group Working Families, said: "The conflict that often seems to exist between family and work is so unnecessary and counterproductive.
"I would particularly encourage men to consider working flexibly as that has the potential to create stronger families and go some way to addressing the inequalities that still exist in the home."
The move is backed by the Confederation of British Industry, but the CBI has warned that firms may find it difficult to grant requests for flexible working during the recession.
Trades Union Congress general secretary Brendan Barber said the extension would be "good news" for employers.
He added: "Companies whose workers enjoy the benefits of flexible working are likely to feel less stressed and less anxious about balancing their lives at home and work, which means they are more committed and productive while at work."
Meanwhile, the new payment for pregnant women - called the Health in Pregnancy Grant - is intended to help expectant mothers stay healthy in the run up to the birth and help meet some of the extra costs.
All mothers-to-be can ask for a claim form from their midwife at their first antenatal appointment after the 25th week of pregnancy.
The grant is payable to all women irrespective of their income. It is not taxable and will not affect other benefits such as tax credits, Income Support or Jobseekers Allowance.
The Thomas family talk about their flexible working hopes
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