His son Richard Kay said his father had thought fees like this were usual practice in the aviation industry.
"Quite extraordinary, but he is 84 and simply must have thought this how it is when re-booking flights," he told the Independent.
Air New Zealand told the Independent the usual "change fee" on the route was around £80 ($174.40) per person.
Mr and Mrs Kay also suffered from further inconveniences when their passports were stolen on their way into the city from Sydney Airport.
They were issued temporary documents by the British Consulate which allowed them to continue on to New Zealand.
However, they were told they would need to obtain new passports in Auckland before travelling back to Australia - which required them to change the date on their flight yet again.
This time, the couple's daughter-in-law Lesley Kay changed the date for them through CheapOair - and was charged a further £950.40 ($2071), which she assumed was the company passing on the airline's fees.
At this point, CheapOair's "Post Booking Charges" had reached an astronomical £1999.40 ($4358) - more than seven times the original cost of the flights.
When their son contacted CheapOair about the fees his parents had incurred, he was told the matter would go to the billings department "for clarity" - and his emails went unanswered.
However, when contacted by the Independent, the agency said the charges were a mistake.
Mr and Mrs Kay were reimbursed £1390 ($3029) which was later increased to £1608 ($3505).
"The delay in resolving this error does not reflect the high standards of service that CheapOair strives to achieve," the company said in a statement to the Independent.
"CheapOair is reviewing its checks and measures to reduce the likelihood of human error during a booking amendment."
The agency is an offshoot of an online travel agent based in the US, with its British base in West London.
A spokeswoman for Air New Zealand told the Herald any fare purchased through an agent or other third party would be subject to the third party's terms and conditions.
"If a customer then elects to change their booking, Air New Zealand's standard change fee is applied along with any difference in fare," she said.
"The third party may also charge additional service fees on top of this which is beyond airlines' control."
A customer would generally be referred back to their original agent or point of sale, as the agent would have visibility and control over their full itinerary, she said.
- nzherald.co.nz