What are the nicknames of American airborne regiments?

Paratroopers.
No. Paratroopers are the name for airborne soldiers of any country who descend by parachute. American airborne regiments probably have their own lesser-known nicknames, and perhaps they can be obtained by researching each regiment individually. I can recall only one offhand. The 555th in World War 2 was nicknamed "the Poor Old 555th" because they were held out of combat throughout the war, and the only action they saw was extinguishing forest fires caused by Japanese balloon bombs. The 555th was as good as any other American airborne regiment, but they were a black unit, and racism was a factor.

I also remember that the 503rd was the regiment which recaptured the island of Corregidor on February 16, 1945. Corregidor was called the ROCK and as a result the regiment is still called the ROCK Regiment. They are currently assigned to the 173rd Airborne Brigade in Vicenza, Italy.

The nickname for the 173rd is the "Sky Soldiers". This nickname was given to them by the Chinese when the 173rd was stationed in Japan and conducted all manner of training jumps into the orient. The Chinese called them "Tien Bien" meaning, "Sky Soldiers".