The government is planning to make the registration of fathers' names on birth certificates compulsory.
Unmarried fathers must register to gain parental responsibility
Ministers said they would push forward with the plans once safeguards to protect vulnerable women and children could be put in place.
The Institute for Public Policy Research said the move would send an important signal about fathers' roles.
Currently a birth has to be registered within 42 days, but only one parent is required to do so.
The Department for Work and Pensions said proposals on joint birth registration in England, Wales and Northern Ireland would be brought forward after a full consultation.
Some people had been expecting the Secretary of State for Work and Pensions, John Hutton, to include the measure in the Child Maintenance Bill.
But it is thought that a rethink was prompted by concerns about women who had been subjected to violence from a partner, and rape victims.
However, a Department for Work and Pensions spokeswoman said it was never intended that the proposal would be in the bill, as birth registration was not in the department's remit.
The government was still committed to the idea, she said.
If the parents are not married, the father has to be registered on the birth certificate in order to gain parental responsibility.
If no father is registered he will be listed as unknown.
The IPPR said fathers had a vital role to play in the life of their children and were more likely to be actively involved if they were there from the outset.
It also said that unless fathers were registered, they could not be traced and encouraged to make child maintenance payments.
Mr Hutton is said to support these arguments.
The institute's head of social policy, Kate Stanley, said: "Everyone should know who to send a card to on Father's Day.
"Most people will be thanking their fathers tomorrow but many will be wondering who their father is and why they have not helped support their family.
"Requiring fathers to be registered on a birth certificate sends an important signal about the duties of parenthood.
"It communicates the message that fathers have an equal role to mothers and that they must take their responsibilities seriously."