Hackett, who has struggled with drug and alcohol problems away from the pool, was detained by Gold Coast police on Wednesday after his parents reported he had suffered a "breakdown" at their family home.
He was released after a few hours without charge but a day later, his parents said he had disappeared.
"Grant's been found alive and sober," Neville Hackett said outside the family's Mermaid Waters home. "He's spoken to police.
"He's actually hiding because he's very, very embarrassed about all this," he said, adding that while he was unsure where his son had been he had made contact with police.
Earlier on Thursday, Hackett's father had put out a plea to find the former Olympic swimmer.
"He's definitely a missing person," he told local media. "He's mentally disturbed and needs urgent help ... If anybody has seen him contact the media or the police or the Hackett family.
"Grant, let us know where you are. We love you and we want to help you."
The swimmer's brother Craig told reporters on Wednesday that Hackett was a "danger to himself and to the community" and that his family were unable to help him alone.
However, before he was reported missing, Hackett posted a photo of himself on Instagram with cuts and bruises to his face and said his brother had beaten him.
"My brother comments to the media... but does anyone know he beat the shit out of me," Hackett wrote. "Everyone knows he is an angry man."
It is unclear when or where the photo was taken and Hackett did not appear to be sporting any injuries when he was released by police.
The incident has shaken Australia's sporting community and the Australian Olympic Committee (AOC) expressed its concern for Hackett.
"This is just a big shock and surprise," AOC President John Coates told Sydney's Daily Telegraph newspaper. "I am very worried for him and his family.
"He is one of ours. He's an Olympian, and one of our greatest Olympians. We'll give him whatever support we can."
'RANTING AND RAVING'
After Hackett's arrest, his father said the 36-year-old was "ranting and raving a bit" and the family had called the police when he had refused to get treatment.
"He's big and powerful when he's not happy," Neville Hackett said. "We decided he needed some treatment but there was no way he was going to go and get treatment this morning, so we called the police."
Regarded as one of the greatest long distance swimmers of all time, Hackett won back-to-back 1,500 metres golds at the 2000 Sydney Olympics and at Athens in 2004 before retiring after the 2008 Games in Beijing, where he won silver in the event.
The 10-times world champion, who was also a member of the 4x200m freestyle relay gold-winning squad in Sydney, returned to the pool last year to make an unsuccessful bid to qualify for the Rio Olympics.
In the wake of national trials, he was involved in an altercation with a passenger on a flight and questioned by police at Melbourne airport.
He said he had been binge drinking after his Olympic disappointment and publicly apologised.
Hackett has previously struggled with addiction to sleep medication and spent time in a rehab centre in 2014 after he was spotted in the lobby of a Melbourne casino disoriented and wearing only his underwear.
(Reporting by Ian Ransom and Sudipto Ganguly; Editing by Peter Rutherford/John O'Brien)
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