|Photo of Muhammad Musallam held by his parents|
IS said Mr Musallam, 19, had posed as a foreign fighter when he joined the militants in Syria. It said he later confessed to being a spy for Israel's Mossad - a charge denied by Israel and by his family.
"Islamic State is doing this because it has an objective: it wants to scare the entire world," said Said Musallam, the man's father.
"How does it want to send that picture to the world - to the entire world and the Arab world? The way is through Muhammad."
He said his son went missing after travelling as a tourist to Turkey in 2014.
A few months later an online magazine linked to IS carried a long interview in his name in which he confessed he had been sent by Israel to infiltrate the organisation. Its publication was a clear signal that he had fallen foul of the group and was in danger, our correspondent Kevin Connelly says.
The 13-minute video purportedly showing Mr Musallam's death has not been independently verified. Israel has declined to comment.
The video shows a boy aged about 12 accompanied by an older, French-speaking militant. He issued threats against Jews in France before the boy shot an orange-suited figure dead.
It is the latest in a series of IS propaganda videos appearing to show the killing of hostages. Last month IS militants in Libya released a video showing them beheading 21 Coptic Christians.
There is no level of depravity too low for this demonic horde. The deceived fools think they are acting in God's will when, in fact, they are doing Satan's bidding, and doing it very well.