As club captain, Davies, 29, will assist City's main skipper, Stephen Darby, with some off-field duties around the club.
Davies made his return to the side for Saturday's win over Crewe Alexandra after missing eight games with a broken arm.
Ricky Ravenhill was the club captain for last season, with Darby taking over the role following the midfielder's departure midway through the campaign.
Parkinson said: "Andrew will work with Darbs (Stephen Darby) to liaise with Steve (Parkin) and myself about things that may crop up around the club.
"It is nice to have more than one senior player to bounce things off.
"Davo (Andrew Davies) is one of our leaders and his presence in and around the dressing room is important.
"We have used club captains before with Ricky Ravenhill helping Gary Jones until he left the club. It is something that works well."