In this video, an american WW2 vet claims he shot down an enemy soldier in a parachute as "retaliation" for the enemy pilots breach of an unspoken rule that said you didn't shoot at parachuters in the air, on the account of them being easy targets.
I am wondering if there is any truth to this, and if so, did this unspoken rule also apply to parachute-soldiers annd not just pilots jumping out of a plane
It depends on when and where in the war you're talking about. Generally, there was some sort of unspoken rule, although it was definitely broken at times. Many airmen talked about letting themselves fall as far down as possible before deploying a parachute for fear of being shot while drifting down. The Polish squadrons in the Battle of Britain had the reputation in the British squadrons for "taking no prisoners" and shooting airmen in their parachutes; I can't remember exactly where I read it, might have been in "Fighter Boys" by Patrick Bishop. I don't know how well earned that reputation was. Of course, the pacific conflict was a much more racial and brutal than that in Western Europe, and several American airmen were reportedly shot by the Japanese and vice versa in the South Pacific. All that said, there are also interviews where pilots said that they would never shoot at somebody who had bailed out, so it goes both ways. Unfortunately, I don't know of any formal study of the prevalence or numbers of this happening, so much of the evidence is anecdotal.
It is worth noting that Article 20 of the Hague Rules of Air Warfare (1923) explicitly prohibits the shooting bailing pilots. It was never adopted, but the fact that this was considered a possible explicit "law of war" is relevant.