TUC General Secretary Frances O'Grady said a "good gift" this Father's Day would be for ministers to increase statutory paternity pay rates and for employers to top it up for longer.
Ms O'Grady said: "Poor levels of financial support are preventing new dads from taking extra time off and are particularly affecting low-paid fathers who simply cannot afford to take leave.
"Extending paternity pay from two to six weeks and paying a better statutory rate would make a massive difference, as has been shown in other countries".
More top news
Crime figures show a sixfold increase in people reporting a rape by someone they met on a dating app or website between 2009 and 2014.
The launch was made in defiance of United Nations sanctions barring North Korea from using ballistic missile technology.
David Cameron has raised concerns after figures showed 100 babies spent time living behind bars last year.