Police said the man arrested was called Alireza Japalaghy but did not name the woman.
According to Tasnim News Agency, officers said in a statement: “Images of a young boy and girl who exhibited improper and unreligious behaviour were published on social media.
“These people were arrested by the police with the judiciary officials’ command, because what they did was a sample of ‘advocating vice’.”
Parkour, which is also referred to as freerunning, involves getting from A to B in the quickest and best-planned way possible. Practitioners vault over walls, railings and other obstacles in acrobatic moves that are popular on social media.
Women’s rights are stringently restricted in Iran and wearing a headscarf is compulsory in public for all women there – with those who do not wear a hijab, or have some of their hair on display while wearing a hijab, facing punishments ranging from fines to imprisonment.
Mr Japalaghy shared multiple photos of the pair kissing and embracing on his Facebook, Twitter and Instagram account alongside the caption “Dawn of Tehran”.
He was arrested on Monday, while the woman was detained on Thursday, according to ABC News.