Dr Parmar said in a statement today she would resign if she won the nomination.
But Mr Prasad said it was morally wrong to be campaigning for National while holding a commissioner position.
"The Families Commission is an autonomous crown entity required to base their findings on evidence and to critique policies and programmes that are inconsistent with that evidence.
"Any perception of a political conflict of interest compromises this role.''
State service guidelines say public servants should remain politically neutral.
It is not the first time National has been accused of conflicts of interest in relation to the Families Commission.
Labour criticised the appointment of chief commissioner Belinda Milnes, who is Cabinet Minister Amy Adam's sister and a former staffer for Social Development Minister Paula Bennett.
Ms Milnes said in a statement today that she had taken advice from the State Services Commission and was confident that the Families Commission's processes were robust.
She said she had made it clear to Dr Parmar that if she won the nomination she would have to stand aside for the duration of the election campaign.
"It is important that public servants and those appointed to Crown Entity boards are able to exercise their rights to stand for political office and I believe that Dr Parmar can manage this in an appropriate manner.''
Ms Milnes said she had listed a conflict of interest in the board register, which meant Dr Parmar could exclude herself from any areas which could have a perceived or actual conflict.