A study, conducted by the Change Bereavement Leave campaign, found seven out of 10 UK employees would back a national guaranteed minimum of paid leave.
While many employers exercise discretion when a close relative dies, only 15% of workers questioned correctly understood there was no guarantee they could take paid time off, the study of more than 1,500 people found.
Lucy Herd, the founder of Change Bereavement Leave, began the campaign in 2010 after her 23-month-old son, Jack, drowned in their garden pond.
Her partner at the time was only allowed to take three days off work by his employer - one of which was for the funeral.
Herd called this an "inhumane anomaly", which meant it was vital the law was changed.
She said: "David Cameron acknowledged he was able to take two weeks off after the death of his own son, but sadly not all parents have sympathetic or understanding employers or can afford unpaid time off.
"We would like to see four weeks of paid bereavement leave for parents."
Last week, speaking on prime minister's question time, Cameron acknowledged it was "an important issue".
In September, Glasgow South MP Tom Harris introduced a Bill in the House of Commons to introduce a national guaranteed minimum bereavement leave. However, without Government support there is little chance of success.
Harris said: "This is an injustice that parliament needs to address. I was pleased that at prime minister's questions on Wednesday, David Cameron responded positively to me and seemed keen to consider a change in the law.
"Working from figures available in the Commons library we have come up with a very conservative estimate that at its highest level the cost of the measure to employers across the UK would be £4 million a year."