Labour Party leader Phil Goff believes it would be "almost automatic" for the head of a government department to lose his or her job if that person no longer had the confidence of the minister.
The position of Corrections Department chief executive Barry Matthews is under the spotlight after a highly critical report by Auditor-General Kevin Brady into the handling of parole slip-ups.
Corrections Minister Judith Collins is refusing to express confidence in Mr Matthews and has given the State Services Commissioner 10 working days to identify who is responsible for the failings.
On Wednesday, Ms Collins told Parliament she had not received a letter of resignation from Mr Matthews.
Mr Goff says it is within the rights of a minister to sack a chief executive if there are good grounds.
Mr Brady criticised the way the department has handled offenders on parole, warning that its practices may have posed a risk to public safety.
Mr Brady reviewed 100 case files, more than half of which were considered to be high risk.
He conducted the review after parolee Graeme Burton murdered farmer Karl Kuchenbecker in 2007.
His report has found that, in most cases, the department had failed to follow one or more of its own requirements on the management of offenders' sentences.
It says Corrections ignored its own safety rules, mismanaged cases and appears to be under-resourced.
Mr Brady's recommendations include requiring the department to ensure offenders are not housed near their victims when placed back in the community.
The department says it has dealt with many of the deficiencies identified in the report.
Ms Collins, who tabled the report in Parliament on Tuesday afternoon, described its contents as "alarming".
The ACT Party says failings within the department have been identified before and Mr Matthews should be held accountable.
The union representing probation officers, the Public Service Association, says problems with the parole service need fixing before new programmes such as boot camps can be introduced.
Association national secretary Richard Wagstaff says high staff turnover within the department need to be dealt with urgently.