Prime Minister Jacinda Ardern has previously said that Adams' idea has merit, and if it comes to fruition, then the Opposition would be given due credit.
But she did not want to have it as part of the Government bill because there were details that needed the scrutiny of a select committee.
Adams said she would move a motion in Parliament tomorrow to have her member's bill introduced.
"I'm hoping all parties will support the bill going to select committee. This approach can be a real win-win for the Parliament."
She said there was a surge of public support for the idea of splitting the paid leave.
"You can think of any number of situations where parents might find it really useful to have the two of them home at one time - if you've had a caesarean section and the mother can't drive for six weeks, if there are twins, if one of the children has a medical condition.
"We're giving them a chance to do the right thing and back it, and allow parents to have that flexibility."
Lees-Galloway said Adams' previous attempt to amend the Government's bill was not workable, "fundamentally flawed", and didn't take into account other changes to the bill that would need to be made.
"I'm not prepared to commit us to any position on a bill we haven't seen, especially when the previous effort from this member hasn't really been up to scratch.
"The National Party seems to be in an awful rush, after nine years of doing absolutely nothing on this issue."
He said Adams had not taken up his offer to work collaboratively on the issue.
"Instead she just slaps demands down on the table and issues a press release. It looks more to me like a political stunt."
Adams said the concept of splitting the paid parental leave was "simple" and, if necessary, her bill could be amended after a select committee process - but Lees-Galloway rejected that.
"I would rather get it as close to right before we go to select committee."
He said the Government would look to introduce a better bill early next year as part of other changes to employment relations law.