Two goals within a minute from Peterborough duo Lee Angol and Chris Forrester shortly before the hour mark proved enough to send City to their first defeat in sixth.
The Bantams could have taken the lead before Peterborough's quick-fire goals when Steven Davies hit the post but Parkinson's men struggled to get back into the game following the visitors' double strike.
Parkinson said: "There wasn't a great deal in the game in the first half.
"Apart from Evo's (Lee Evans) set-plays, the quality of our crosses into the box wasn't good enough. The one decent cross we did put in from open play, Hans (James Hanson) has troubled their 'keeper with a header.
"We had a great chance to go ahead at the start of the second half but we've then conceded two quick goals.
"That knocked us clearly, as you may expect, but the disappointment from us all is there wasn't enough urgency to get back in the game.
"There was almost an acceptance from us that today wasn't going to be our day.
"We didn't quicken the game up enough. They were a tiring team towards the end and it wouldn't have taken too much drive to get back in the game.
"The intensity levels we played with against Sheffield United last weekend weren't there.
"Sometimes when you close down with intensity, you get the ball back to you in good positions and you are switched on to use it in the right way.
"There wasn't the precision in our passing from last weekend and there is no reason why that should be the case."
Although with the blow of losing the game, City also had to ensure the loss of Paul Anderson to a nasty looking injury in the first period.
Anderson was forced off with a potentially serious leg injury following a crunching 50-50 tackle with Peterborough's Michael Bostwick.
Parkinson added: "We shall have to wait and see with Paul but he is in hospital having an x-ray.
"It is a blow to us because Ando (Paul Anderson) has been playing well recently.
"His injury probably did knock us as well today (Saturday), perhaps too much.
"It is no disrespect to Davo (Steven Davies) who came on in his place, but we did miss him.
"Steve has played wide in his career, and I just felt he might get us a goal with him cutting inside on his left foot or arriving at the back stick from one of Marshy's (Mark Marshall) crosses.
"We have to regroup now and make sure we go to Colchester (United) on Tuesday and put in a stronger performance to earn a result."