The France No9 ruptured his posterior cruciate ligament in the 19-10 defeat to South Africa in Paris on Saturday and faces up to three months on the sidelines as a result.
That length of absence would make him unavailable until mid-February, ruling him out of all four fixtures Clermont have left in Pool 4, starting with the double header with the Scarlets in a fortnight's time and ending with games at Harlequins and at home to Racing Metro in mid-January.
Clermont have confirmed that the 25-year-old will be absent until the knockout stages, although they insist the setback isn't quite as serious as some first feared.
"Morgan was injured late in the game between France and the Springboks at the Stade de France on Saturday and has since had an MRI scan," read a statement on the Clermont website.
"The scan revealed a rupture of the posterior cruciate ligament in his left knee. The injury does not require surgery but will rule him out of action for between 10 and 12 weeks."
Parra played in all nine games of his club's run to a first Heineken Cup showpiece last term, starting every tie and contributing 113 points as Clermont beat Leinster, Exeter Chiefs, Scarlets, Montpellier and Munster before being narrowly beaten by Toulon in Dublin.
Those stats surpassed his efforts the previous year, when he again started every game and scored 56 points as Clermont reached the semi-finals, where they were beaten by champions-in-waiting Leinster in Bordeaux.
He started this season's opening Heineken Cup fixture at Racing Metro, kicking all of his side's points as Clermont went down 13-9, but he sat out the 23-16 win over Quins after being banned for four weeks for punching in a Top 14 clash with Bordeaux-Begles.