Clarke picked up the problem with his knee during the first period of the Scunthorpe game - an injury which was later aggravated thanks to a crunching tackle from visiting midfielder Isaiah Osbourne after the restart.
The Bantams are set to send Clarke for a scan to find out the exact level of damage to the Irishman's knee but he is certainly set to miss the Boxing Day trip to Fleetwood Town and the home game against Notts County two days later.
Clarke, who is City's joint top scorer after scoring three goals in his last four games, is also likely to be sidelined for the F.A Cup tie at Millwall in early January as well.
With City seven games unbeaten since Clarke's sizable impact as a half-time substitute against FC Halifax Town last month, Parkinson has acknowledged that he may have to look again at how his side will shape up for the visit to Fleetwood.
Parkinson said: "It looks like Billy Clarke will be out for a few weeks. We are pretty sure of that now but we will be sending him for a scan to be sure.
"Billy had a slight problem with his medial ligament in the first half (of the Scunthorpe game). It wasn't major but the tackle by (Isaiah) Osbourne on the halfway in the second half, which I felt was worthy of a red card, has rattled Billy's knee a bit.
"I think with the adrenaline of the game, Billy was able to carry on. Once that has wore off, however, the extent of the injury has surfaced. It is a little bit worse than what we first thought.
"As a result, we will need to find a balance in the team that is right and allows us to continue to get points on the board.
"The team in recent weeks has had a nice feel to it, and that has been key to us getting some good results lately.
"Billy playing in that sort of 'number ten' role has been an important part of our recent success and we have to decide now whether we stick with the same side and go with a Mark Yeates or Billy Knott in that role, or whether we go back to a more conventional 4-4-2 formation.
"We will be looking at a few things in training to see what we think will work best."
Along with the news about Clarke's injury, Parkinson also confirmed that, despite taking a knock to his shoulder against Scunthorpe, Portuguese winger Filipe Morais should be fine for Boxing Day.
In other injury-related news, City midfielder Matthew Dolan is now back at the club receiving treatment after injuring his hamstring in the warm-up last Saturday while on loan at Hartlepool United.