A 53rd-minute strike from Emile Sinclair proved enough to give the Red Devils maximum points and manager Barker was quick to make the point that this was the type of game his side would not have won last season.
Parkinson said his team paid the penalty for being "too open" as they slipped to their second successive 1-0 reverse.
"We were too open as a team," Parkinson said. "When your formation is very attack-minded away from home you have to play to the letter, and we didn't do that.
"We had a lot of chances and had our spells in the game, but were too open as a team and that left us vulnerable."
Midfielder Garry Thompson wasted two opportunities inside the opening 30 minutes and Parkinson added: "We had enough moments to get something out of the game, but were too expansive.
"Crawley are a side with good movement and we gave them too much room when we lost the ball.
"We've lost our last two games 1-0, but both games could have gone either way. When you have spells you have to score.
"There was a lot of dejection in the dressing room and we know there are areas where we can get better."
Parkinson, expecting a reaction when the Bantams visit Preston on Tuesday, added: "It's a great stadium and it would have been nice to go there with something from this match.
"But we know we will need to work really hard on Tuesday."
- Sports & Recreation
- Emile Sinclair