A single strike from former City trainee Emile Sinclair early into the second half was enough to settle a contest which saw promising scoring chances on offer for both teams.
And while Parkinson's side had a fair share of opportunities fall their way, the Bantams chief was disappointed that City allowed hosts Crawley too much time and space to engineer openings of their own.
Parkinson said: "We've had chances to score today (Saturday), particularly in the first half. When you have that sort of pressure though, you have to really score. We didn't and that has left us open to being punished.
"Last twenty minutes or so, we've also had some good chances to get a goal. With that in mind, some supporters might leave the ground and feel we were unlucky not to score and that we could have got something from the game.
"My opinion is that I felt we were too open as a team. Obviously I pick the team, and we have been looking to go with an attack-minded team away from home recently. If your attacking players don't play to their best when it comes to performing their defensive duties though, the pitch suddenly becomes a very big place.
"Unfortunately I felt vulnerable when we didn't have the ball.
"We can't deny that they have had some good scoring chances as much as we have.
"I just didn't feel today that the discipline in the team when we didn't have the ball was to the standards we expect.
"We have gone away from home recently to Gillingham and Walsall and tried to play like the home team.
"We did that because the lads thrive on that sort of high-intensity football that you usually associate with a home team. Throughout the majority of my time here, pretty much all of our away wins have been because of attacking football. We haven't had too many backs-to-the-wall wins.
"But when you pick so many attacking players away from home, you are relying on them to do the job for the team. I just felt we had too many players not quite at it when it came to their defensive roles and that gave the opposition too much room to work with on the pitch."
As they did against Tranmere Rovers last weekend, City went to three at the back in the game's closing stages in an attempt to salvage something from their visit to the Checkatrade.com Stadium.
The move nearly worked in injury time but Rory McArdle, who returned to the side after international duty, was denied when his header from a Raffaele De Vita corner was repelled by a smart reaction save from Paul Jones.
Parkinson added: "Credit to the lads, they did keep going to the end. We all jumped up when Rory's had his header at the end - we thought it was going in.
"We kept going, we kept trying to unlock the door. Hanson has won a lot of headers for us but did we get on them quick enough? Did we anticipate them well enough? The answer is no.
"For all the aerial challenges he won, especially in the second half, we probably didn't get enough for it."
Partnering McArdle in City's central defence at Crawley was debutant Matthew Bates.
The former Middlesbrough man lasted 76 minutes on his first league start since playing for Bristol City back in April before he was sacrificed to allow the Bantams to shift into their 3-4-1-2 shape.
Parkinson continued:"It was good for Matthew to be involved.
"He has obviously been waiting ten days for his chance since signing. He is a good player, he just needs games.
"I thought he was one of our better players. He just tired towards the end a bit, which we expected given his recent lack of games, so it was a pretty easy decision to bring Matthew off when we decided to got to three at the back because we wanted to protect him."