Second best for large parts of a one-sided tie at Victoria Park, a City team featuring as many as six changes from Saturday's draw at Stevenage were brushed aside with worrying ease by Colin Cooper's Pools in the first round encounter.
Although alterations had been made to City's starting eleven for the match, with the likes of Gary Jones, Andrew Davies, Nahki Wells and James Hanson not even making it onto the pitch, Parkinson was quick to point out that he had still expected much, much more from the players that were selected.
And despite City's confident start to their League 1 campaign, Parkinson wasn't content just to sweep this result under the carpet and class it as 'only' a cup defeat.
Clearly concerned by the nature of this humiliating reversal, Parkinson said that poor performances like the one served up by his players at Hartlepool were simply not acceptable at all.
Parkinson said: "That certainly wasn't a performance here tonight (Tuesday) that represented the club of Bradford City, myself, Steve (Parkin) or the fans.
"Since I've been here, we have always made changes in the Johnstone's Paint Trophy, yet we've also always had a spirited performance from the players selected.
"During the first twenty minutes of the game, I thought that would be the same with this game. We've conceded the (opening) goal though and there was no sort of response from the players.
"Looking at cup ties for ourselves in the past, yes, we have conceded first in some of those matches, but we have always fought back and tried to get back in the match.
"But tonight we just simply didn't defend properly. We didn't win the first headers, we didn't contest 'keeper's kicks, we didn't defend the centre of our goal at all at any point in the night, and we got punished for it.
"I have to look for reasons why that was because that is not a performance I want to see again.
"There was a feeling of acceptance out on the pitch. There seemed to be no urgency to want to respond once we had conceded. I don't ever remember seeing that before here and I don't want to see that sort of attitude again.
"We have worked hard to build up a reputation around the football club and around football in general of being a side that is full of honesty and commitment, but you can quickly lose all of that if you play we have just done.
"Forgetting for one moment that this was a cup tie, if some of the players we have here are happy to be an also-ran in this division (League 1) and think that promotion into League 1 is the pinnacle of their career, then they are not the right players for Bradford City.
"I need players here who are hungry and want to improve. We always play with hunger, but too many people were second best tonight."
Parkinson continued: "I've selected players tonight to give them the opportunity to impress and stake a claim. When they get given a chance, however, they have got to play in a manner that says to me 'I should be in the team'.
"Their performance should speak a thousand words. The talking shouldn't be done in my office or in the training ground, it needs to be done on the pitch.
"Credit must go to Hartlepool, of course, they played very well tonight, but my main thoughts are with us.
"Whoever is picked on Saturday against Brentford has to leave the pitch having made sure we have put in a lot better performance than we did here.
"Whether we win, lose or draw against Brentford, as long as we play in a manner befitting Bradford City then I'll be fine with that."
As a result of City's shocking exit out of the Johnstone's Paint Trophy, Parkinson's men will be back in training this Wednesday as they prepare their clash with promotion hopefuls Brentford at the Coral Windows Stadium in Sky Bet League 1.
Wednesdays are usually reserved as a day off for City's squad, especially after a Tuesday night match, but Parkinson has cancelled the club's normal plans and brought his players in to immediately begin work on making sure the woeful Hartlepool display is not repeated any time soon.
Parkinson added: "You can't simply walk into the dressing room after losing 5-0 and say 'right lads, grab your stuff, we'll see you Thursday'.
"This is our job - we get paid for this. We get paid to try and bring joy and entertainment to our supporters, who pay their hard-earned money to watch us play.
"Our supporters have seen us lose games quite heavily in the past but they have probably still seen effort and commitment in our play and clapped us off the pitch. They understand when they see a team giving it everything.
"But tonight, they didn't see enough and I didn't see enough.
"We have to make sure that, yes, we draw a line through this game, but that we also learn a lesson from the match as well. The main lesson is that if you don't do the basic rights, you don't win your tackles and your headers, then you won't win games of football."