Fathers should be given a month off work after the birth of a child and £200 a week paternity pay, the Equal Opportunities Commission has said.
Cards calling for "Daddy's month" are being sent to party leaders
An EOC poll of 1,200 fathers found a third were not happy with the amount of parental leave they had and 80% would be happy to care for the baby at home.
It is delivering Father's Day cards to politicians including Tony Blair calling for a "Daddy's month" of leave.
However, the CBI said extending paid leave would be "a step too far".
While unpaid parental leave is available to men, the EOC wants at least a month at a guaranteed £200 a week - almost double the £106 men are paid for the two weeks of paternity leave they are currently entitled to.
EOC chief executive Caroline Slocock told BBC News: "There has been a dramatic shift in attitudes by fathers towards caring for their new born babies.
"Nine out of 10 men with babies now agree that they are as confident as their partners in caring for their new children."
She said this was a major shift from 20 years ago, when 52% of fathers believed it was their responsibility to be the main breadwinner while women stayed at home to look after their children.
Ms Slocock said: "We want policy to catch up with the modern dad.
"What we want to see, as a Fathers' Day present for all new fathers, is better paternity leave and the ability for mothers and fathers to share parental leave over the next six months of their child's life."
She said one of the reasons for the change in fathers' attitudes was the narrowing of the income gap between men and women.
The survey showed 32% of men were earning equal or less than their female partners before their child was born.
Duncan Fisher, chief executive of campaign group Fathers Direct, said the EOC research showed how much new fathers wanted to be involved with their babies.
"Our leave system will continue to be antiquated and out of step with modern families unless the government revises its plans by creating extended, well-paid paternity leave and introduces shared parental leave rather than transferable maternity leave," he added.
However, the CBI rejected the idea. "We need to maintain a balance between parental desire for flexibility and employers' need to get the work done," a spokeswoman said.