Most parachute/glider troops carried M1 rifles, not carbines. In general, carbines were issued to company grade officers, millitary police and those whose principal weapon was not a rifle. The soldier in the photo cited is armed with a 57mm recoilless rifle; the M1A1 Carbine would get in his way less than a rifle or a full length carbine.
In the older (pre-1941) TO&E, such troops would have been armed with a pistol or revolver; the Carbine was adopted because it was felt that it would be more effective than a pistol without the weight and burden of a rifle.
Yup. My uncle carried one in the war. He was in the 505th PIR, 82nd Airborne div. He was a machine gunner.