A single Lee Miller goal fourteen minutes in helped the hosts crucially edge a tight contest in their favour.
As they had done last weekend at home to Crewe Alexandra, City dominated proceedings after the break at a chilly Brunton Park. This time, however, Parkinson's men couldn't find a breakthrough to ensure that Miller's opener gave Carlisle a barely-deserved victory.
Parkinson said: "We've started the game well enough but then conceded a sloppy goal from a set-play.
"We had at least a couple of chances to clear the ball but we couldn't.
"I felt we lost our way a bit after the goal up until half-time really.
"We lost our composure out on the pitch.
"It wasn't anything to do with a lack of commitment, we just lost our composure and started slashing at things, playing the wrong ball and generally not thinking clearly.
"I felt we controlled things a lot better in the second half though.
"We dictated the game after the break for large periods and some of the football we played was good - we just lacked that moment in and around the box.
"Goals are so important in games and this is not the first time we've conceded the first goal in a match.
"We couldn't get our goal when we were on top in the second half. Goals change games and we know that."
City defender Rory McArdle gave a perfect example of the commitment and desire that is going to be required by Parkinson's squad to end their current winless run by playing at Carlisle despite still sporting a nasty gash under his left eye.
Parkinson added: "In difficult conditions, I can't fault the commitment and effort of the back four and of the team as a whole in the second half.
"Rory McArdle was a good example of that. He has a nasty gash under his eye and medical advice was telling us probably not to play him tonight, but he put himself up to play.
"In the situation we find ourselves in at the moment, I think it would be very easy to just hide in the treatment room.
"Rory certainly didn't though, he's put his hand up to play tonight (Tuesday) and put in a great performance."
In City's quest to get back into the match, Parkinson took off recent loan signing Matthew Dolan at half time and also replaced James Hanson with Andy Gray with ten minutes to go.
Parkinson confirmed after the game that neither substitution was to do with any fresh injury concerns for either player.
The Bantams chief said: "With Matty, I just felt he looked a bit leggy. It was a big ask of him to play again after Saturday.
"I thought we would need (Nathan) Doyley's freshness and quality on the ball in the second half, and he certainly showed that.
"With regards Hanse (James Hanson), it just seemed he was struggling a bit towards the end of the game. He's had a bit of a hamstring problem recently and hasn't trained too much in the last week or so as a result.
"I also felt Andy (Gray) might just give us a bit more craft in the final minutes of the game."
On a generally forgettable night for City, their main standout performer in the game was probably winger Adam Reach.
Reach was a constant threat to Carlisle's back four throughout the match and looked City's most potent attacking weapon for large parts of the fixture.
Parkinson said: "He was magnificent at times tonight.
"He tormented both full backs - he's had a good chance for himself in the first half and created a couple others in the second.
"He's such a great athlete, he can cover the ground so quickly. He looked a real threat when we got him the ball."