He explained on Twitter that his “mandate was one of progression” and felt he could not continue following defeat of the proposal to increase promotion into the league.
Member clubs voted 9-7 for no change rather than consider options to increase the number of teams who are relegated.
One team currently go down, to be replaced by the winner of a three-way play-off between the champions of the East, West and South of Scotland Leagues.
Parkinson said: “Following the defeat of the proposal of increased promotion into the SLFL I informed the chairman and secretary post meeting of my decision to resign by position on the SLFL board with immediate effect.
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“My mandate was one of progression, and a 7/9 is progress of sorts, given I was told privately not to even bother with it a couple of seasons ago such was the likelihood of change happening.
“Clubs are absolutely entitled to vote for what they believe is best for their club; democracy has to be abided.
“However given the status quo has been voted for I feel I no longer have a mandate to serve on the board and will focus my time on preparing for our impending play-offs.”
Bonnyrigg will be home to the Highland League champions in the first leg of the pyramid play-off on April 23, with the return leg the following Saturday on April 30.
The winners will face the club who finish bottom of League 2 over the two subsequent Saturdays.