Shortly before Kyle Vassell's winner for Peterborough, Davies had to leave the field after colliding with City goalkeeper Jordan Pickford as the pair tangled for a cross into the box.
Davies required nearly three minutes of on-field treatment after the collision with Pickford, something that contributed to a mammoth total of nine minutes of injury time being added on at the end of the match.
The defender is set to go for an x-ray to determine the exact nature of the damage but Parkinson is confident City have the players in their ranks to cope should Davies' injury prove to be as bad as initially feared.
Parkinson said: "Unfortunately for Andrew, it is a suspected broken arm.
"He will be going to hospital for an x-ray though, so we will wait for that before saying for certain.
"Andrew is a good player for us but let's get one thing straight, if Andrew comes out of the team, we have enough players within this squad to fill his shoes.
"Hopefully he is not out for too long and he is back with us soon. If he is out for a while, however, there will be no excuses from us. You lose players to injury throughout the season and you have to adapt to that, and we have enough players within the squad to do that."