Clarke has missed City's last five matches with a knee injury but he has been making significant progress on the training ground over the last week.
Parkinson will make a late call on whether the striker will be included in his travelling party for the Yeovil fixture.
After sitting out Wednesday's F.A Cup win over Millwall, Jordan Pickford will definitely be back in City's squad for the game. Matthew Dolan, who spent a period earlier in his career on loan at Yeovil, could also make up part of the group after recovering from a recent hamstring problem.
Parkinson said: "Billy Clarke is working really hard on the training ground and he is getting ever closer all the time.
"It was a tremendous night on Wednesday but everything is concentrated now on Yeovil.
"We will be travelling on the way down so we can prepare as thoroughly as possible given the time restraints.
"We know we are up against a side who are fighting for their lives and we need to focus our minds on the job in hand."