Immigration New Zealand confirmed to Cooke's parents that he had entered the country. However, he hasn't been seen or heard from since he landed.
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Cooke had travelled extensively internationally and within Australia and always touched base to let his parents know what he was doing when he arrived somewhere, his father John said.
"But we've heard absolutely no news from the 8th [of January]... He's arrived in Auckland and that's all we know."
It was unclear where in New Zealand Cooke was travelling to, but it was likely he had come to sell strawberries, a job he had been doing in Australia for the last six to nine months.
He had never been to New Zealand before, John said, but was supposed to be meeting up with a friend when he arrived.
Police on Friday appealed for any sightings of Cooke, saying they were continiuing to make enquiries and would like to hear from anyone who may have seen him.
Cooke is about 167cm (5ft 6) tall with blue eyes.
He always dresses entirely in plain black clothes and often wears a hat and sunglasses, his father said.
Anyone who has seen Cooke or has any information about his whereabouts is urged to contact police immediately.
- additional reporting: Sarah Harris