There is anger among parents at minister Regina Doherty's suggestion that money is not the reason for the low paternity leave take-up rate among fathers.
Concerns have been also raised about the impact on both public and private sector employers of government plans to extend parental leave.
Ms Doherty has committed to giving new mothers and fathers two paid weeks off from November, rising to seven weeks by 2021.
The State will pay €245-a-week to parents which may be voluntarily topped up by private companies if the employee is on a higher salary.
However, a recent survey found almost two-thirds are not providing salary top-ups. Separate research shows 60pc of men don't take existing paternity benefits.
Earlier this week Ms Doherty suggested there is "a narrative that the value of money in the scheme isn't enough for men to take off work".
"It doesn't seem to have stopped women from taking maternity leave," she added.
Laura Erskine, the head of community for parenting website MummyPages.ie, said that the feedback they have been getting is that Ms Doherty's remarks are "downright insulting".
"A huge proportion of dads simply cannot afford to avail of this leave despite the fact they desperately want to spend this important time with their new or growing family," she said.
Ms Erskine said the government's plans for extending parental leave are welcome but that if the payment that comes with it is not topped up by employers it can be difficult for both parents to take it.
Ms Erksine suggested the government should incentivise employers to top up the basic €245 State-funded paternity and parental leave payments.
She said an "appropriate tax break" would "mark a positive step towards greater balance of parental care of children under a year in the home."
Separately, Mary Connaughton, director of the Chartered Institute of Personnel and Development (CIPD) warned of the pressure on both public and private employers to pay top-ups for extended parental leave.
Ms Connaghton said there will be an "immediate cost" to the State paying to boost parental leave top-ups for public sector employees like teachers and civil servants.
This echoes concerns by Education Minister Joe McHugh about the financial costs of extending the leave.
Ms Connaughton also said the tight labour market will see pressure on private sector employers to pay the top-ups which she said would be particularly difficult for small business.
She argued that there's a "real risk" of a divide between the public and private sectors with State employees being paid the top-ups with others missing out because small businesses can't afford it.
Ms Doherty added that she never claimed new fathers don't want to spend time with their children and acknowledged that most do want to do this.
However she has spoken of the "need to have a conversation in this country on gender roles".
She said the minimum wage introduction was said to be unaffordable for business but it's now accepted that it's a benchmark for fair play in the workplace.