Clark disclosed his displeasure with Linington of the Isle of Wight following a second-half incident concerning his captain Paul Robinson.
Clarke revealed: "There was an incident involving (Burnley striker) Danny Ings and Paul Robinson. I thought it was a foul by Ings.
"They scored from the resultant free-kick. Paul remonstrated with the referee, who gave Paul a yellow card, but he mocked him about the scoreline at the time. This is totally unacceptable.
"When my captain is fighting for the club and the supporters are backing my team and when I am working as hard as I can to get the results to remain in the league, getting mocked by an official is something I am not going to accept.
"I was furious and I lost my temper.
"I am not questioning his performance. But he knows what he said and it is not acceptable in any walk of life.
"There seemed to be a bit of an arrogance among the officials and that is not being accepted.
"This was the only downside, and I have to flag it up. It is that it is not acceptable for an official to mock any player at any stage."
Clark feels that the incident took the gloss off a tremendous fightback by his side after they trailed three times against second-placed Burnley to earn a valuable point.
The Clarets increased their unbeaten run to seven games to maintain their second in the table position but they failed to produce the knock out punch.
St. Andrew's is not their favourite ground as Burnley have now failed to beat Birmingham on their last five visits to the Midlands.
It appeared as if they were about to end their jinx after Dean Marney opened the scoring the first half. But Federico Macheda equalised in the 64th minute only for Michael Duff to regain the lead for Burnley three minutes later.
But Emyr Huws crashed home a 25-yard volley in the 70th minute with Sam Vokes netting in the 86th minute only for Macheda to obtain a well merited last-minute equaliser as a reward for Birmingham's tenacity.
Clark admitted: "It was an exciting hard game against a really good team who were always looking to win.
"For us to have performed as we did and get a point in the manner we did I'm absolutely delighted.
"If we had come off the pitch losing the game it would have been criminal not to have got something for their efforts. We were brave and created a lot of chances."
Burnley boss Sean Dyche had mixed feelings about the result.
He said: "It was fantastic point. "We came off a massive win against Blackburn.
"People must have wondered if we could again against Birmingham.
"Do they have that resilience? Do they have that belief? I thought it was there for all to see as we took the lead three times.
"The fitness of the players was superb. The will and demand was there. We didn't play as well as we can but we came away with a valuable point and that is a great sign of how we have played this season."