Mark also appears to say in his speech that the $25,000 in Government funding No Duff received last year was "courtesy of me".
National Party defence spokesman Mark Mitchell said Mark's comments were indefensible.
"This is a Cabinet Minister standing up in front of a voluntary organisation and basically threatening them. It's not like he's at a NZ First conference or something like that. He is at a public event as a minister.
"He spoke as if it's his money - $25,000 - and then says it's very apparent that they're not supporting NZ First.
"The Prime Minister needs to start taking this seriously. This is not the kind of behaviour New Zealanders expect of their ministers."
Mark told the Herald that he wasn't threatening to cut funding to No Duff.
"The assertions that I campaigned for New Zealand First or threatened to cut funding to No Duff in my speech lack context and are incorrect," Mark said in a statement.
"If someone has taken what I said that way, they are mistaken. My speech that day was 25 minutes long. The comments that are being joined together happened a long way apart, and are unrelated."
On it's Facebook page, No Duff said it was an apolitical organisation.
"We hold no view on the Minister's recent comments, but we appreciate his willingness to speak frankly and with candor.
"The Minister is an important member of the Veterans community. He is one of us."
Last week co-founder of Animal Evac NZ Steve Glassey accused Agriculture Minister Damien O'Connor of threatening to cut Government funding, which O'Connor has denied.
"We have had veiled threats from officials and even a minister that if we continue to draw attention to [Ministry of Primary Industries'] deficiencies, our chances of getting funding will be affected," Glassey told a parliamentary select committee.
He accused O'Connor of saying to him: "You can't go riding us and then come to us for funding."
But O'Connor denied this, saying that Glassey was always negative about the great work MPI did.
"Steve will always extrapolate things out. I said it's really important to be positive when we're trying to negotiate a better deal with him,:" O'Connor said.
"I think Steve going around criticising MPI staff at every single opportunity when everyone is doing their best is not a very productive way forward."
When asked about O'Connor last week, Prime Minister Jacinda Ardern said that Ministers should not threaten to cut Government funding and she was open to seeing any evidence of such behaviour.