Parkinson beat off competition from Reading's Steve Clarke, Aston Villa boss Tim Sherwood and Arsenal chief Arsene Wenger to pick up the honour.
The LMA is the official organisation for all professional football managers - the League Managers Association.
The City boss was chosen for the award after leading the Bantams to an FA Cup quarter-final replay after memorable victories over Millwall, Chelsea and Sunderland in earlier rounds.
The FA Cup manager of the year award is judged by the LMA board and goes to the manager whose team has delivered achievement beyond expectation in the competition, either with a significant performance or progress.
The award is the second to be given out by the LMA to Parkinson during his time as City boss after he received the LMA's Special Merit Award following the club's memorable 2012/2013 campaign.
AFC Bournemouth's Eddie Howe was given the main LMA Manager of the Year title this Tuesday while Bristol City's Steve Cotterill won the League 1 managerial award.
Wycombe Wanderers' Gareth Ainsworth won the League 2 Manager of the Year award.