If Mr Parker did resign, it would be a step that would effectively be a vote of no confidence in current leader David Cunliffe and a sign Mr Parker either intends to stand himself or back another candidate for the role.
Mr Parker said this morning he was still the deputy leader of Labour after yesterday's seven-hour caucus meeting but would not comment further when asked if he was likely to step down.
"That's the only comment I'm making."
Mr Parker has not ruled out a tilt himself although he has done so in the past partly for personal reasons.
Mr Parker has stayed quiet since election night. He stood beside Mr Cunliffe for his morning press conference yesterday but was not there when Mr Cunliffe spoke to media after the caucus meeting.
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During the morning press conference Mr Cunliffe interrupted when Mr Parker was asked a question, saying Mr Parker could not answer any questions.
Mr Parker also said he did not intend to answer questions - but he did intervene at one point to reject a suggestion that the caucus had undermined Mr Cunliffe since he became leader.
Mr Parker said that had not happened.
Mr Parker had refused to reveal which of the three candidates he supported in the last run-off between Grant Robertson, David Cunliffe and Shane Jones.
He has been Mr Cunliffe's deputy ever since, although he is not considered a Cunliffe loyalist.
He told the Herald earlier this year that he had taken that stance because of a need to bring caucus together, saying it was clear bridges needed to be built and he was one of those in a position to do so.
Mr Parker was going on a break yesterday, saying he was going skiing.