A complaint has been made to police about a threat to ACT's new deputy leader Beth Houlbrooke after she made controversial remarks about families.
A baby holds a parent's hand.
Ms Houlbrooke posted a screenshot of the alleged abuse on her public Facebook page. The message said she needed a "bashing or a bullet".
Police have received a complaint and are making inquiries into the matter, a spokeswoman says.
Last week Ms Houlbrooke sparked a furore when she said families who can't afford children shouldn't be having them.
It was said in the context of criticism of Labour's $60-a-week baby bonus policy.
She said Labour was paying people to have babies, which encourages them to have children when they're not mature enough or financially prepared.
"Labour's baby bonus could extend the misery of child poverty and even child abuse," she said in a Facebook post on Wednesday.
"The fact is, parents who cannot afford to have children should not be having them. ACT believes in personal responsibility, meaning we stand with the majority of parents who wait and save before having children."