New Zealand First wants the minister responsible for Novopay, Steven Joyce, sacked over the service's latest blunder.
The salary details of hundreds of education staff were briefly made available for staff at other schools to see, the latest in a series of botch-ups with the system.
"This is yet another Novopay shambles that puts individuals' privacy at risk and puts more strain on school staff at a stressful time of year," says New Zealand Educational Institute (NZEI) national secretary Paul Goulter.
NZ First education spokeswoman Tracey Martin wants Novopay scrapped and a whole new payroll system started from scratch.
"I'd like to see somebody fired over this, because a huge number of New Zealanders have had their pay, and now their privacy, stuffed up. What does it take for a minister to get fired?"
Ms Martin says Mr Joyce was supposed to be the system's 'Mr Fixit', but instead he's made it worse.
Meanwhile Mr Joyce has promised a "full investigation" by the company into how the error happened.
"It is particularly frustrating as it comes after a long period of stability in the payroll service," he said on Thursday.
Novopay was beset with problems from its launch in mid-2012 and in 2014, the Ministry of Education took over responsibility for it from developers Talent2.
A review of the system last year showed its error rate was down to 0.2 percent, within the 0.5 percent "acceptable" rate, and complaints were down 81 percent on 2014.
But it hasn't all been plain sailing. In April the NZEI claimed Novopay cut wages for school support staff, when it divided their annual pay by 27 fortnights instead of 26. Mr Joyce said at the time no one would be underpaid.
Ms Martin says while Novopay was built in Australia, she wants a New Zealand company to design a new payroll system.
"The public servants that are trying to work with Novopay were handed a dog. They're going to have to build a completely new system because this one just has too many patches on it," she says.
"I really think they have to go to a New Zealand company that understands the New Zealand system and set up a reliable program."