Davies was forced to come off after only 17 minutes during City's match with Oldham Athletic on Saturday after appearing to damage his hamstring chasing visiting striker Jonathan Forte to a forward pass.
The influential centre half will be sent for a scan to try and find out full story regarding the injury but Parkinson already believes Davies will be looking at certainly a 'few weeks' out of action.
Davies' absence from City's line up means Gary MacKenzie is set to start against Chesterfield at the Coral Windows Stadium on Tuesday night.
Parkinson said: "It hopefully doesn't seem too bad for Andrew but it looks like he'll be out for a few weeks.
"(Head Physio Matt Barrass) is going to send Andrew for a scan today (Monday) to enable us to get a proper look at the injury.
"It is disappointing for him clearly but, looking at the incident, he appears only to be running at three-quarter pace when it happened so it doesn't look too bad.
"Matt is not overly concerned either but, equally, any hamstring strain, even if it is a minor one, will keep a player out for a few weeks.
"Gary (MacKenzie) has been excellent for us when he come into the side though.
"Having not played a first team game all season before coming here, Gary is growing into his role and getting physically stronger all the time.
"His body is once again adapting to the rigours and demands of first team football. He has fitted in well with the group and hopefully he can continue to play as well as he is."
With Davies definitely out of Tuesday's match against Chesterfield, Parkinson admits midfielder Andy Halliday will probably also miss out due to a minor hamstring problem.
Parkinson added: "Andy has just got a slight issue with his hamstring.
"I don't think it is too bad - we are sending him to a specialist to probably have an injection in there.
"Once he's had that injection, we're confident that will clear it up.
"Whether he makes Tuesday, we're not too sure, but he'll definitely make Friday (vs Doncaster Rovers)."