The hosts controlled proceedings for long periods of the contest and created numerous chances throughout.
They ended the match having had eight shots on target, with Wrexham failing to test goalkeeper James Beadle.
Birmingham’s xG (expected goals) of 1.55 dwarfed Wrexham’s meagre figure of 0.08 while the Blues had almost four times as many touches in the opposition box (22) as Wrexham (six) managed.
It meant the Red Dragons failed to score in an away league match for the first time in 11 games while they tasted defeat on the road in the Championship for just a second time in 10 outings.
“First half we didn’t connect the game in terms of the top third of the pitch,” said Parkinson.
“I thought we switched it well. We got Issa [Kabore] in some great positions. We grew into the game in terms of the football.
“But it’s just then, in that area, can we pick the right pass, show that element of calmness to then make sure that top third possession results in chances.
“In every game this season, we’ve carried a threat, we’ve been a goal threat, even against Premier League teams in the cup.
“We’ve played top teams in this division, we’ve scored goals and today we just couldn’t find a way to produce that quality.”
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