Davies had to leave the game ten minutes into the second period following a clash with City goalkeeper Jordan Pickford as the pair both went to claim a cross into the Bantams box.
With the bone displaced following the accidental collision with Pickford, Davies could require surgery to have the arm plated.
An exact length of time Davies is expected to be sidelined for is not completely known as of yet but the defender could be facing roughly two months out due to the break.
Parkinson said: "It is clearly a blow to lose Andrew - he is probably fitter than he has even been at the moment.
"He has had a great start to the season but we have the players within the squad to cope. We can't use this injury as an excuse.
"It is up to the player who comes into the side to replace Andrew to keep hold of the shirt now. We will regroup and come back strong."