Parkinson's men were unable to stop their hosts from making it thirteen games unbeaten in all competitions after a lacklustre first half showing at SportsDirect.com Park.
A close range finish from Oldham top scorer Jonathan Forte and a deflected drive from outside the box by Mike Jones saw Lee Johnson's Latics deservedly two goals in front with less than half an hour gone.
A first goal for the club from Andy Halliday on the stroke of half time gave City an unlikely lifeline though, and the Bantams actually looked more like their usual selves in the second period.
With Billy Clarke, Mark Yeates and Gary Liddle all enjoying promising moments in the Oldham box after the restart, Parkinson's men could have even snatched something from the game.
Despite City's improved showing in the second half though, Parkinson wasn't ready to forget the Bantams' off-colour opening period performance.
Parkinson said: "We were second best in the first period.
"We sat down with the squad yesterday (Friday) and showed them some footage of Oldham and how they start games.
"They caught Coventry City on the back foot here on Tuesday night and we said that couldn't happen to us, but we did exactly that.
"The players have got to look at themselves and I've got to look at myself because this group of players have gone out and done exactly the opposite of what we've asked them to do.
"Physically, we weren't strong enough in the first half and that resulted in us being second best in our individual battles across the pitch.
"It could have been more than two goals for them if I'm being honest but we got the goal back through Andy Halliday and then looked a lot better in the second half. We looked the most threatening team in that half.
"The main issue to come from the game for me, however, is that we were nowhere near good enough in the first half and that simply isn't good enough.
"I fully understand that the fans will be annoyed with the team and myself because this was a Lancashire-Yorkshire derby match and we've come out second best.
"You can't always pass it the way you want to do in football but you always have to stand up and be counted as individuals and we didn't do that today.
"We must take some credit for the second half display because we could still have taken something from the match. We got in some great positions but I just didn't feel there was enough conviction in our play or in the final ball.
"Billy Clarke has had a couple of good chances, Lids (Gary Liddle) has found himself in a good position, (Mark) Yeatesy has also had a couple of chances, but when you get those chances when you are two-one down away from home, you need someone to take them.
"All in all then, it has been a very bad day at the office for us and we need to have a long think about what we are going to do in training this week because too many times recently we haven't had a performance which is going to get us the result."
During a difficult opening period for the Bantams, Parkinson took off Frenchman Christopher Routis for James Meredith six minutes before the interval.
Parkinson confirmed after the game that the move, which saw Alan Sheehan move across to centre back, was a tactical one and Routis hadn't suffered an injury.
Parkinson added: "I thought Chris was poor today (Saturday). He got dominated by (Oldham striker, Jabo) Ibehre.
"I have to be fair to him, he has done well since he came into the club and he is still learning the language and our English culture.
"We had spoken to him beforehand about the intensity of games in England and the start that is required.
"Chris impressed in the MK Dons league game where it was two teams playing a lot of possession football.
"Today was different, however, they (Oldham) got it quickly into Ibehre and the two centre backs didn't cope."