Tickets went on general sale for the fight at Cardiff's Principality Stadium at midnight on Tuesday (NZT) and within five hours there were only a limited number of $571 and $1142 tickets remaining.
Promoter Eddie Hearn revealed on his Twitter account that the three cheapest categories had sold out in just over an hour after they had gone on sale.
The March 31 fight is sure to sell out with the stadium holding 78,000 under its retractable roof.
Welsh media reported that the sales were a record for the massive stadium that holds rugby and football matches as well as concerts.
Joshua fought Carlos Takam at the venue last October with over 70,000 tickets being sold in the first eight hours.
The quick sales fully justify the insistence of Parker's promoters at Duco Boxing that this bout could be a huge drawcard with fight fans.
Hearn had initially felt Joshua v Parker would be best suited to London's O2 Arena which holds around 20,000. He questioned Parker's drawpower.
Parker's promoter David Higgins successfully argued that with three of the four heavyweight belts on the line, the fight needed a stadium setting.
The on-selling market was quickly into gear and making some huge profits.
The cheapest tickets at NZ$76 were being sold for between $129 and $156.
VIP ringside tickets which originally cost $3800 were being onsold for as much as $11,455.
The All Blacks sold out their test against Wales last year at the same venue two months ahead of the November fixture with a 74,500 capacity for the rugby configuration.
As the rush to get the boxing tickets left some fans disappointed, Joshua moved quickly to help one of the contractors painting the canvas floor for the ring that will be used.
Carl Jones had tweeted his plight and Joshua responded quickly by offering him two tickets.
Earlier, UK police had warned about fake tickets being sold on the internet as fraudsters looked to cash in on the huge occasion.