Although suffering an early injury in the 2011 World Cup final in Auckland, the scrum-half had been a key figure in their run at the tournament, and the 24-year-old was expected to again add versatility on next month's tour.
But the Clermont number nine aggravated his thigh during his side's Top 14 semi-final defeat to Castres, forcing him to withdraw from the national squad.
"That is stopping him being at 100 percent, notably in his place kicking," a France team spokesman said of the injury.
"Following consultation with the Clermont medical staff and those of the French team, he must pull out of June's tournament in New Zealand."
Half-back Jean-Marc Doussain has been announced as the man to take Parra's place in the squad - the Toulouse man's only Blues appearance coming as a late replacement in the 2011 World Cup final.
"Jean-Marc Doussain was the player on the stand-by list, it's logical that he joins us," added France head coach Philippe Saint-Andre.
The Blues first face the world champions in Auckland on June 8 before further tests in Christchurch and North Plymouth.