Parkinson, who had been at the helm for just over three years, and his assistant Steve Parkin have quit as off-field problems continue to dog them.
Bolton, who started the campaign on minus 12 points and postponed their game with Doncaster on Tuesday night, say they are considering the resignations.
A spokesperson for Bolton told the PA news agency: "The club has received the notices of resignation from manager Phil Parkinson and assistant manager Steve Parkin and the matter is being considered."
Wanderers are awaiting the completion of a takeover by Football Ventures which had been blocked by a court order issued by Laurence Bassini, who had tried to buy the club earlier this year.
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And as the issue rumbles on, Wanderers have been hit on the pitch as they have been forced to field a team full of youngsters in their first few games.
Bolton aren't the only League One club in trouble with Bury facing expulsion from the Football League on Friday unless they can prove they can satisfy financial obligations this season.
The Shakers are yet to play a game this season after all of their fixtures to date have been suspended.
If they are kicked out then the league would consist of just 23 teams this season with three relegation places - instead of the usual four.