The race is now between Mr Shearer and a yet-unnamed deputy candidate, and David Cunliffe and Nanaia Mahuta.
Mr Cunliffe, the current finance spokesman, has identified himself with the centre-left of the party over recent days, while Mr Parker is seen as being more to the right.
People committed to Mr Parker had not automatically switched their allegiance to Mr Shearer, making the leadership campaign closely contested.
The Otago Daily Times understands deputy leader Annette King and campaign manager Trevor Mallard may have reacted to Mr Parker's poor performance on television on Wednesday night and had decided he was not viable as a leader.
Mr Mallard oversaw the general election campaign just past.
Labour Party general secretary Chris Flatt yesterday released a timetable which showed the leadership contenders would be in Dunedin next Friday to meet local party members and explain their credentials.
The road-show starts on Monday in Hamilton, followed by Palmerston North, Wellington, Christchurch, Dunedin and ending in Auckland on Sunday, December 11.
The meetings were open to party members only. The formal leadership vote would be undertaken by caucus members on December 13.
However, given the election result gave Labour its worst result in 84 years, it is likely the feelings of members will be conveyed to their 34 MPs during the consultation process.