On top of the mounting problems for France, Parra's Clermont teammate Wesley Fofana is also likely to be ruled out for the rest of the Six Nations after coming off during the 20-13 defeat against Wales on Saturday.
Fofana is out for up to three weeks with a torn thigh muscle.
The Paris-born back, 27, who operates at centre or wing, made his international debut in 2012 for France and has been at Clermont since 2008, helping the side win the French championship in 2010.
Parra was forced off in the 53rd minute of France's defeat to Wales, their second loss in three matches, and underwent tests on Monday which revealed the extent of the injury.
He will now miss the rest of France's Six Nations campaign and Clermont's European Champions Cup quarter-final against Northampton on April 4.
Fofana could be back for the Saints clash although Clermont's Portuguese flanker Julien Bardy is a serious doubt after he came off while on international duty with a thigh problem against Germany and is set to be out of action for a month.
France opened their Six Nations campaign with a win over Scotland but have since suffered defeats in Ireland and at home to Wales.
Their final two matches are against Italy and England.
Ireland are the only unbeaten side with three wins with England and Wales having picked up two wins in three outings followed by France and Italy on one victory and Scotland at the bottom having failed to win a point.