The rescue service said the Israeli military attacked an ambulance and a fire engine in the southern Gaza Strip city of Rafah on March 23.
The bodies of 14 of the men were recovered from a mass grave seven days later.
The Israeli military said at the time that its forces had opened fire on several vehicles after they approached troops in a suspicious manner, without co-ordination or headlights.
Israeli officials said the soldiers had killed members of the Hamas and Palestinian Islamic Jihad groups.
The Palestinian Red Crescent said on Saturday that it had retrieved a mobile phone from one of the killed paramedics, which held video material of the rescue squad.
The video, shared by the rescue service on X, shows clearly marked ambulances and a fire engine driving with their headlights and blue emergency lights switched on.
The video recording stops after less than a minute when the convoy comes under Israeli fire.
However, the audio recording continues for several minutes and includes prayers spoken by the attackers as well as unintelligible shouts of command in Hebrew by the Israeli soldiers.
The Palestinian Red Crescent said the unarmed rescue workers were shot at close range, contrary to the Israeli account that soldiers fired at approaching suspicious vehicles.
"This video unequivocally refutes the occupation's claims that Israeli forces did not randomly target ambulances, and that some vehicles had approached 'suspiciously without lights or emergency markings'," the rescue service wrote on X.
An Israeli military spokesman told the Times of Israel newspaper that the incident would be thoroughly investigated, including the video and audio material.
