The managers clashed on the touchline just minutes before the final whistle after a foul by Wednesday's Scottish international Barry Bannan on substitute Filipe Morais.
Referee David Webb gave Bannan a yellow card for the incident, but then dismissed the rival bosses after they squared off in the technical area.
Security then needed to intervene as Wednesday manager Carvalhal appeared to tussle with a steward.
"Bannan, for me, has made a dangerous tackle right in front of us," said Parkinson after the game. "I have left the technical area by a couple of yards to have a go at him.
"Maybe I shouldn't do that but it is a natural reaction. Carlos has run from their dugout right across to get involved. After that the ref sent us to the stand."
Carvalhal admitted he deserved to be sent to the stand after the confrontation with his opposite number.
"It is easy to explain," he said. "There was a foul from [Bannan] and I didn't like it because [Parkinson] pressured the referee.
"I went to the fourth official and said he can't press the referee. With the emotion of the moment I was out of my technical area.
"I think the decision of the referee was correct. But there are reasons why I moved from my position."
After being embroiled in a tussle with the steward, Carvalhal added: "I don't want to talk too much about that.
"He apologised to me. It was a person with age. I must respect older persons but in that moment he acted really badly."
Watch the above video to see both managers tussle with each other on the touchline.
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