Sharapova, who was the 2004 Wimbledon champion, exited the match while trailing 5-0 in the third set. She previously served for the match while leading 5-3 in the second set but eventually lost to Parmentier 4-6, 7-6 (4), 5-0.
Sharapova, who was unseeded at Wimbledon for the first time since her debut at the event in 2003, called for a medical timeout following the second set. She had her wrist taped by a trainer and play resumed.
The Russian star stopped play again in the third set and retired after Parmentier served for the 5-0 lead.
"I've had a history of a tendon in my left forearm flaring up," Sharapova told reporters. "It happened today in the second set. I probably couldn't have gotten through the whole final set, but I did my best."
Sharapova's participation during this year's Wimbledon was only her second tournament since January after she underwent shoulder surgery. She fell in the second round in Mallorca last month.
In other women's action, Serena Williams earned a straight-sets win over Italy's Giulia Gatto-Monticone 6-2, 7-5.
The seven-time Wimbledon champion, who is dealing with an injured left knee, is attempting to win her 24th career Grand Slam title. She takes on Slovenia's Kaja Juvan in the second round.
Federer rallied after losing the first set against South Africa's Lloyd Harris, earning a 3-6, 6-1, 6-2, 6-2 win at Centre Court. Federer plays Britain's Jay Clarke in the second round.
Nadal, playing in his first match since winning his 12th French Open title, defeated Japanese qualifier Yuichi Sugita 6-3, 6-1, 6-3 to reach the next round. The Spanish star faces Australia's Nick Kyrgios in the second stage of the tournament.