The visitors took the lead through Martin Skrtel's own goal after 20 minutes, but Liverpool turned things around by finding the net twice in quick succession midway through the second half.
Daniel Agger and then Daniel Sturridge converted almost identical Steven Gerrard set-pieces before Dowd sent off Shola Ameobi for dissent in the wake of what turned out to be the winning goal.
Substitute Paul Dummett also saw red shortly before the end for a challenge on Luis Suarez, with Pardew left frustrated by Dowd's display.
"The referee played a part in my view," he said. "I think he should have managed the game a lot better than he did.
"We conceded two goals - I felt he had a part in the first one, I didn't think it was a foul by Vurnon (Anita). The second one I felt we certainly should have had a foul by Agger on Shola."
Pardew was especially disappointed about the dismissal of Ameobi and felt it was harsh on a player probably playing his last game for the club.
"I know Shola," he said. "Shola doesn't swear and I saw the referee and said 'well how angry can he possibly get?'
"I think what happened was he (Dowd) kind of said 'if you carry on I'll send you off' and maybe sarcastically Shola says 'well send me off', and he did.
"That's sort of the explanation Phil gave me. I think he should have managed that far better than that.
"It is disappointing for Shola because we assume it might be his last game and I think he was terrific in everything he did.
"It was a shame that that's happened to him, and if the ref had handled that better he'd have finished the game like he should have.
"It's very, very disappointing for him and for our fans because he's one of the finest players I've managed."
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