"Parra has suffered a ruptured posterior cruciate ligament in his right knee," his Top 14 club Clermont said in a statement.
He started Saturday's 20-13 home defeat by Wales in Paris and was injured in the 53rd minute.
Parra will also miss Clermont's European Champions Cup quarter-final against Northampton on April 4.
Parra's absence means France coach Philippe Saint-Andre is likely to field Sebastien Tillous-Borde alongside flyhalf Camille Lopez when Les Bleus travel to Italy on March 15.
The two started all three tests in November before Toulon's Tillous-Borde missed the first two Six Nations games through injury and started on the bench on Saturday.
France are fourth in the table after they opened their Six Nations campaign with a 15-8 win over Scotland before losing 18-11 in Ireland and being defeated by Wales for the fourth time in a row.
Saint-Andre is expected to name a 30-man squad to prepare for the Italy game later this week.
(Reporting by Julien Pretot, editing by Ed Osmond)
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