This is an issue where the government had hoped to reach an agreement across the political divide. But unless they can also get the Left Party on board, the proposal will not get through parliament. It illustrates the tricky situation that the red-green government is in, after the election.
The Liberal party, which normally acts as part of the opposition four-party Alliance, has signalled that they are prepared to vote in favour of a proposal to reserve a third month in the parental leave for the dad.
"If the proposal only includes this, we will vote in favour. We think there ought to be one more earmarked month in the parental leave," Liberal Party leader Jan Björklund told Swedish Radio News.
At the moment, two of the 16 months of parental leave are reserved for either parent. This means two months that only the dad can take out, two months that only the mother can take out, while the remaining 12 months can be divided according to what suits the family best.
But usually, it is the mother who takes out the majority of the parental leave, which is why it is often said the system contains two "daddy's months". The government, and the Liberal Party - as a way to encourage more dads to take a bigger part of the parental leave - want to increase this to three months, that can not be transferred to the other parent. If either parent does not want to take out any parental leave, that part of the parental leave is then forfeited.
The Left Party wants to go even further. They want to split the parental leave in half, and reserve one part for the dad and the other part for the mother, with no option for one part to take out much more than the other parent.
"This is what is necessary to sort out the big differences in income, salary and pension between men and women," said Hans Linde, who is the leader of the Left Party group in parliament.
The Green Party, which is part of the government, wants to reserve a third of the parental leave days for either parent, and then a third that each family can divide up how it suits them best. But they see a third "daddy's month" as a step in the right direction and is therefore behind such a proposal from the government.
The other three parties in the Alliance, ie the Moderates, the Christian Democrats and the Centre Party, are against increasing the number of months reserved for either parent. This is also the position of the Sweden Democrats.
This means that if the government does not get the support of the Left Party on this issue, they will have trouble in getting it through parliament. And Hans Linde is clear that the government cannot take their support for granted.
"The government has not contacted us about this proposal. We will not give our tacit support to a government proposal unless we have been involved and negotiated about it," Linde tells Swedish Radio News.
The government proposal on the parental leave is expected in May, to be decided on by parliament during the summer.