After opening the scoring against Shrewsbury Town last Saturday late in the first half, Clarke was forced off midway into the second period after falling awkwardly during a City attack.
Following a scan on the injury this Monday, the Bantams have now received the unfortunate news that the Irish forward could be ruled out until the month of November.
Speaking after the club's 2-1 loss to Gillingham, Parkinson said: "Billy will be out for 10-12 weeks.
"It is a huge blow for us because he is a player of real quality.
"His quality was really missed tonight (Tuesday, vs Gillingham). We needed someone to get hold of the ball and to keep it, like he did on Saturday (vs Shrewsbury Town).
"We have had injuries to players before, however, and we have dealt with them.
"The squad is good enough but we are not showing it at the moment.
"I have to find the answers as to why this is the case, and I need to find them quick. I am confident we will do."
Frenchman Christopher Routis came off with a tight hamstring during the first half of Tuesday's Gillingham match. Parkinson is hopeful the injury isn't too severe though.
Parkinson added: "Chris is a player who had played every minute for us so far before the injury. He is a great athlete but his hamstring tighten up and we had to bring him off.
"If you look around the country, a lot of players are picking up strains at this time of year.
"When you play all of the games so far like Chris in a short space of time, that does leave you susceptible to this sort of injury.
"We hope it isn't too serious, however, and that he is back sooner rather than later."