The Norton Fulbright survey looked at how more than 200 employers are getting ready for the legislation and their attitudes to the changes.
Two-thirds said they are worried about the increased administrative burden they will face.
Just under half of employers (46%) reported they will only be offering the statutory minimum extra leave to both mothers and fathers going forward, with 30% saying they will not offer any enhanced paternal pay.
Paul Griffin, head of employment and labour, London at Norton Rose Fulbright, put the unwillingness to prepare down the predicted low level of take-up from staff.
"The Government’s own estimations of the take up of the shared leave period is expected to be between only 2-8%," he said.
Despite this, Griffin warns that employers should take action to put systems in place "sooner rather than later".
Working Families policy and research manager Jonathan Swan told HR magazine the changes will be most effective in companies where flexibility is already embedded into the culture.
"They will find that shared parental leave complements their existing policies and activities, which probably already have senior buy-in."
