Reiss and Afshar were among more than 10 other defendants brought before the court for a mass hearing on charges related to huge protests that erupted across Iran after Mahmoud Ahmadinejad, the Iranian president, was declared winner of the June 12 election.
Reiss' father, Remi, said: "She does not have any involvement in the region ... Of course she is innocent."
"She has nothing to reproach herself for and nobody can reproach her about anything."
Afshar's son said his mother had been coerced into making false confessions.
"They forced a confession out of her. She is not an activist, this is not a political person at all," Naiman said.
"Her computer was confiscated two weeks ago and she was arrested by intelligence agents on the eve of her departure to France. They lured her out of the door by saying they had come to return the computer."
Naimian said he hoped his 50-year-old mother, a secretary in the cultural section of the French embassy, would be freed due to French pressure, stressing that she had taken French nationality.
The British government and the EU presidency also condemned Iran for putting Reiss and other Iranian workers at their embassies on trial.
David Miliband, the British foreign secretary, said the trial in Tehran was the "latest Iranian provocation", while Sweden, which currently holds the rotating EU presidency, said it was an act against the entire 27-nation bloc.
Miliband said he was "deeply concerned by the unjustified charges" laid against Hossein Rassam, the British embassy's chief political analyst.
"Hossein is a member of our embassy staff going about his legitimate duties. Iranian action against him, and those against Clotilde Reiss and a member of the staff of the French embassy in Tehran only brings further discredit on the Iranian regime," he said.
Iranian news agencies reported that Rassam was accused of spying.
Rassam was one of nine locally employed staff at the British embassy arrested soon after the election. Eight were freed, while he was released on bail.
Rassam, Afshar and Reiss appeared in a group trial with opposition supporters who challenged Ahmadinejad's re-election.
"The EU will closely follow the trial and demand that the persons will be released promptly"
Swedish EU presidency
Iranian state television said Reiss and Afshar had fuelled the unrest by supplying information to foreign embassies.
The Swedish EU presidency said in a statement: "Actions against one EU country, citizen or embassy staff is considered an action against all of the EU, and will be treated accordingly.
"The EU will closely follow the trial and demand that the persons will be released promptly."
Iran has strongly accused Britain of stoking the post-election demonstrations of supporters of defeated opposition challenger Mir Hossein Mousavi.
Britain vehemently denies the claims.