City fell behind after seven minutes, Jonathan Forte given the freedom of the penalty area to turn and fire home.
Mike Jones was then allowed to run from midfield unhindered and pick his spot from 20 yards in the 24th minute to double the home side’s advantage.
Parkinson said: “Too many players were not strong enough. We sat down with the squad on Friday and showed them clips of Oldham’s last two home games when they started quickly and played with real intensity.
“Coventry in particular got caught cold and I said it was really important it didn’t happen to us but their start has won them the game.
“It is not often I can say we have been physically second best. You can’t always pass the ball and sometimes you are not allowed to away from home because of the intensity of the way the opposition play. What you can do – and it is an absolute given – is stand up and be strong in every challenge.”
Andy Halliday pulled one back for City on the stroke of half-time, firing home from James Hanson’s nod down.
But, while they did improve, they never really got going in the second half either and Jordan Pickford made a stunning save from Jones to deny Oldham a third.
City’s woes were compounded in the 91st minute when Rory McArdle was sent off for a second bookable offence
Oldham Athletic: Rachubka; Kusunga, J Wilson, Dieng, Mills; Winchester (Dayton 73), Kelly, Jones, Philliskirk; Forte (Poleon 76), Ibehre. Unused substitutes: Tyson, Brown, Mellor, Turner, B Wilson.
Bradford City: Pickford; Darby, McArdle, Routis (Meredith39), Sheehan; Halliday (Clarke 73), Kennedy (Knott 73), Liddle, Yeates; Stead, Hanson. Subs: Dolan, Mclean, Morais, Williams.
Referee: David Coote (Notts).
Man of the match: Andy Halliday.