City defeated County 1-0 thanks to Billy Knott's fourth goal of the season to extend their unbeaten run to nine games, but Davies only lasted 14 minutes into the contest.
The defender was replaced by Alan Sheehan and City will now have to look at the problem ahead of next Saturday's trip to Championship side Millwall in the F.A Cup third round.
With Davies potentially facing a spell on the sidelines though, Parkinson was quick to point out that City do have cover in central defence, in particular Sheehan and Frenchman Christopher Routis.
Parkinson said: "We hope it isn't anything too serious but we do have good players to replace Andrew if we need to.
"We have shown that before already this season when Andrew broke his arm.
"We have Chris Routis, who is exceptionally unlucky not to have played more so far this season, and we also have Alan (Sheehan) who we went with today (Sunday).
"He was excellent today. When called upon in the second half, he won some big headers and got himself in the right position.
"It is never easy coming on as a substitute, especially in such a key position, but Alan was on top form when he came into the back four.
"As much as we are disappointed to lose Dava (Andrew Davies), it is how we react to it that is important."
Going into their trip to The Den next weekend, City will also be forced to check on the availability of Mark Yeates after the Irish attacker missed the County win following an injury picked up during the Boxing Day victory at Fleetwood Town.
Yeates' fitness wasn't helped by being on the receiving end of a hefty tackle from Fleetwood's Mark Roberts during Friday's encounter, and Parkinson says he remains upset by some of the treatment his players received during that fixture.
Parkinson said: "I am still disappointed by some of the challenges that went in on our lads during that particular game.
"I have spoken to David Allison (Referee's chief for the Football League) about that because the one on James Meredith (by Conor McLaughlin) in particular was a disgraceful challenge.
"The one on Mark Yeates is the one that has injured him. How bad it is we don't know because Matt (Barrass, Head Physio) wants to let the swelling go down before we decide whether or not we need to scan the injury.
"Let's just hope that, fingers crossed, the damage isn't too bad."