508th Parachute Infantry was independent (non-divisional). Attached to 82d Airborne Division January 1944 and remained so throughout the remainder of the war except for a brief period during the Battle of the Bulge (during which it was attached to 7th Armored Division). Participated in the Normandy Invasion, attached to the 82d - not the 101st - Airborne Division. Became an element of 508th Airborne Regimental Combat Team (RCT) in 1948. Perhaps this is the unit you're looking for. (508th RCT had its own shoulder sleeve insignia [patch], similar to that of the 82d Airborne Division, from 1948 to about 1955. If you recall the movie, "Jumping Jacks", with Dean Martin and Jerry Lewis, they wore the patch of the 508th).
If this is the unit you're looking for, I can get you a regimental history ("yearbook") that covers the regiment's activities from 1942 to VE-Day.