Parkinson made eight changes to the side which lost to Oxford at the weekend and the Bantams saw their Johnstone's Paint Trophy campaign ended at the Northern Section semi-final stage.
The Railwaymen scored two late goals in the victory, which Parkinson stressed made the scoreline misleading.
"I thought the lads did really well against a really good team with a totally changed line-up," he said.
"We took them all the way and the scoreline that will be in the papers is in no way a reflection of the game.
"We nearly got back into it but we got caught on the counter-attack but that is the risk you take.
"Our two centre-backs did really well for two young lads on a difficult surface.
"The surface was as frozen as I have seen and if both managers were honest we would have said that game shouldn't have been played."
The Capital One Cup is now City's last chance of a Wembley cup final this season, with their second-leg trip to Villa Park taking place on Tuesday January 22nd.
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