Claiming the U.S. was able to track the parachutists on radar, is where the story starts falling apart. If the Americans were tracking the parachutists as they were coming down then they were tracking their plane. The USAF would be extremely interested in who was flying it.
If the SAS "regularly tested US security", then the U.S. would have known because we are allies. There would have (should have?) been procedures in place for such tests. The Americans who caught the SAS would not have needed 18 hours to figure out who they were. Unless they were testing security without telling American military at all, which seems unlikely.
Finally, playing a practical joke like this during the Cold War would be a bit much, even for the British sense of humor. Or am I just expecting too much common sense?
I agree with the other posters who say it’s baloney.