Remember, the .30cal BAR was designed as a 'trench gun' to move with the infantry, providing automatic weapons support. The damn thing weighed about 20lbs empty. I shot one once way back when...very effective weapon but heavy and an ammo hog.
I don't think the paras carried them due to weight. Instead, I think you will see they used a lot more M1 carbines w/folding stocks, M3 .45cal 'grease' guns, and lots of .45cal Thompsons...in addition of course to the Garands and .30cal BMGs.
Light fighters...not all that heavily armed...just like today.
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