A first half Rams goal that looked offside and the winner from a controversially awarded penalty - both from assistant Mr. Barker's decisions - were among the incidents that provoked City manager Parkinson's reaction.
He has until 30th August to respond, but he sees a silver lining if he can raise awareness of the officials' performances to the FA.
He said: "When the dust has settled a bit, you look back and think maybe I should have kept my mouth shut and gone in the dressing room.
"Unfortunately sometimes the passion and the frustration boils over so I'll take the punishment on the chin. I'll also be asking the FA if the linesman's performance will be reviewed, and if they do that and the assessors look at the linesman's performance in that game, then I'm quite happy to take my punishment on the chin."