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Nicholas Heyward, 31, was killed in the city of Mendoza last Tuesday.
He had been on a rock-climbing holiday through Chile, Argentina and Peru.
Today, his father Ben Heyward told Radio New Zealand how he came to learn that his son had been killed.
"It was a case of the media got to the news before it got to me," Mr Heyward told Morning Report.
"I very quickly realised that something was wrong, and then the journalist realised I didn't know and he backed off.
"It was the New Zealand High Commission in Canberra who said to me that something had happened, but they couldn't confirm anything until Interpol spoke to us.
"For hours, it didn't hit me, it was something I knew in my head, but did not want to believe in my heart. It was dreadful."
Two men have been detained over the killing of a New Zealand man in a park in Argentina, as the friend who was with him talks of how she tried to help him survive.
The first man police arrested was released after eyewitnesses - friends of Mr Heyward's failed to pick him out of an identity parade.
Mr Heyward, who lives in Adelaide, said his son's funeral will be held there next Monday after the body arrives back from Buenos Aires this week.
The family, and his son's large circle of friends, have already started to gather in Adelaide, he told Radio New Zealand.
He remembers his son, a trained physiotherapist, as a "very active child - quick in mind and body".
"[He was] a desperate junior footballer, I mean, he wanted that ball - and that describes how he lived life."
He grew up in Papua New Guinea before moving to Australia where he became a keen surfer.
Latterly, he had toured the world rock climbing which made contact with his father "irregular but frequent".
Mr Heyward had made a comment on his son's Facebook page just an hour or two before he was told he was dead.
"I'd printed off a wonderful photograph of him high up on a cliff somewhere with his back to the cliff, surveying a valley in front of him filled with cloud over the peaks," he said.
"He was greatly enjoying the scenery of Chile and Patagonia and looking forward to climbing in Argentina.
"He was with friends that he was enjoying very much... not telling us a great deal of detail - the photos were doing most of the talking."
While he had a "reasonably clear picture" of what happened, he still wants to find out "a little bit more" and how the police responded.
He has left the investigation in the hands of police and now hopes they catch "these two young blokes with guns".