Their son, Saman Bigy, says every business has a bad day and promises all money will be refunded.
Detective Senior Sergeant Aaron Pascoe said police began looking into the issue last Wednesday.
"It is up to the police to determine if this is a business that has gone bust ... or is it criminal conduct," he said.
Mr Pascoe said the business had stopped operating on Thursday.
When the Herald visited the stores yesterday, a group of people had gathered, claiming money transferred had not arrived.
Siamak Farahi showed the Herald documents suggesting he was owed more than $110,000 that was transferred from Iran in June. It was from his father to pay off his student loan and to start a new life with his family.
"They made up all sorts of stories ... They said the money was lost in Dubai and lost in other ways," he said.
Pedram Noorbakhsk, a 19-year-old student, was unable to pay his rent and had no money for groceries after $3000 transferred from his parents in Iran did not show up last week.
Victoria Williams said she had given the business $12,000 on Sunday to be transferred to Iran, and now feared the money was gone.
In August, Auckland engineer Masoud Moghadassi spent $2850 booking flights to Iran and Malaysia over Christmas and New Year, but his tickets have not been delivered.
When he phoned Malaysia Airlines, he was told the tickets had been booked but cancelled soon after.
Saman Bigy told the Herald he is the manager and director of Dena Travel and that about 150 people were owed money by the businesses, which had had "a bad day".
He said Dena sold travel deals and then booked tickets through The Travel Brokers, which is operated by Stella Travel Services.
Dena had been operating for eight years but ran into trouble when its licence was cancelled last week because it owed around $50,000 to The Travel Brokers. "We were paying off our debt ... This is a problem that has suddenly occurred.
"Any business has good days and bad days, regardless of our business practice. Regardless, we are the ones affected, not the people. They were still travelling on the cheapest deals.
" ... you can't have always the good days but it doesn't mean just because you are having a bad day that you are a criminal."
Mr Bigy said customers could get refunds through Stella. "Everyone is going to get their money back," he said. "Many people have already been refunded as we speak."
Stella chief executive Greg Leighton said Dena was an independently owned member of its Travel Brokers business unit. "We terminated their membership on December 2 ... when we spotted some irregularities in the operation of that business.
"They are certainly responsible for all their commercial transactions. They deal with all of the customer sales, bookings and travel products."
Mr Leighton said it appeared customers who booked with Dena Travel as far back as June or July had been affected and more than 50 people had complained, with 30 refunded.
Persian Network is a registered Financial Service Provider that acts as an intermediary to allow money transfers to Iran, where financial sanctions are tight.
Mr Bigy said his father, Vahid, ran the firm and was unavailable to comment until later this week.
