Blues broke the deadlock on the stroke of half-time, unmarked centre-back Curtis Davies rising to meet Chris Burke's corner for his sixth goal of the season.
Nikola Zigic wasted a great chance to double the advantage immediately after the restart and Peter Lovenkrands hit the inside of a post before Pool's Kirk Broadfoot emulated Davies' effort from Neal Eardley's 74th-minute set-piece.
It was left for Birmingham substitute Wade Elliott to waste a glorious opportunity in the 87th minute as he somehow blazed over from six yards with the goal at his mercy.
Clark said: "It was definitely (two points dropped), I think we deserved to win the game.
"It was a nervy start for the first 15-20 minutes but we grew into the game and finished the half really strongly, we could have been more than one-nil up.
"Then we started the second half really well too, unfortunately we couldn't get that decisive second goal and then they score a very similar goal to us from a set play.
"It's a bit frustrating really. We've played with a lot of desire, tenacity and at a high tempo."
Regarding Elliott's miss, Clark added: "I haven't seen a replay yet but one of the coaches has and it looks like it's bobbled up a bit."
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- Curtis Davies