After coming on as a second half substitute at Bloomfield Road, Davies was forced off in the final few minutes of City's win at Blackpool on Saturday after catching his knee in the turf while turning.
The incident resulted in Davies leaving the ground with a brace on his right knee.
Reece Burke (twisted ankle) and Wes Thomas (calf) also picked up injuries during the match at the weekend but Parkinson says the issue with Davies' knee is the one currently causing him the most concern.
Parkinson said: "Steve is the most serious of those three.
"He is going for a scan this (Monday) afternoon.
"There is swelling around his knee so all we can do is keep our fingers crossed that the injury isn't too severe.
"We are not overly-concerned with Reece and his ankle but he is a doubt for Tuesday night.
"We will monitor Wes over the next twenty-four hours to see if he has any chance of featuring.
"Even if those two can't play though, we do have players ready to come back into the side.
"You look at Nathan Clarke as an example, he has come into the team quite often over the course of this season.
"He adds his leadership qualities to the back four. If Nathan does get the nod, he'll be champing at the bit to play."
In other news, Parkinson says Paul Anderson could be in line to make his long-awaited return to action next Monday when the club's reserves play at Burton Albion.
Parkinson said: "We will monitor Paul this week to see if he has a chance of taking part in that game. Filipe (Morais) is a little bit behind him at this stage.
"Looking at Ando (Paul Anderson) in training, there is nothing to suggest he won't be ready for that reserve game.
"If that is the case, it will be a great boost for him to get some game-time."