Clarke was initially thought to have little chance of playing again this season after fractured his cheekbone against Chesterfield last Sunday.
The Irishman had the injury operated on this Tuesday and City are now hopeful of getting the striker back on the pitch sooner rather than later as long as there isn't a significant risk of further damage.
Parkinson said: "As soon as he (Billy) came round from the op, he was talking about getting a mask so he could play.
"We are looking at getting a mask for him but we had to wait for the swelling to go down first.
"Billy saw a specialist with Matt (Barrass, Head Physio) yesterday (Thursday) but we will ultimately go with the advice of the surgeon.
"Billy is desperate to be involved. He wanted to be out on the training pitch today but obviously we couldn't allow him to possibly receive another bang on the cheekbone.
"He is a good player for us and he wants to make a contribution to these Play-Offs."
While this Sunday's semi-final first leg with Millwall might come too soon for Clarke, Parkinson says City will continue to monitor the fitness of Reece Burke and James Hanson to see if they have any chance of playing.
Burke was substituted at half time against Chesterfield last weekend after tweaking his groin, while Hanson missed the game altogether due to a calf problem.
Parkinson added: "James hasn't trained yet but we are continued to assess him. It is the same with Reece as well really, we are monitoring him too.
"We have other players ready to play if either of those two don't make it - we won't put anyone at risk.
"We won't do that - players have to be 100% to play in this game."
Steven Davies is available once to Parkinson after missing the last three games of City's regular season due to a straight red card at Coventry City last month.
The striker reported a stiff neck this (Friday) morning but he is expected to be fit for Sunday's match.
Parkinson continued: "Steven has a crick in his neck and missed training today to see a chiropractor.
"He has got us some important points with his goals from the bench this season and he has probably been unlucky not to start more games.
"He has been working hard on the training ground incase we got into the Play-Offs and he was needed again this season. Now we are here, he is desperate to play his part."