Parallel reels are simply mistakes, the result of the RSCDS misinterpreting the notes of "dead" dances that they resurrected - Cadgers, for example. All of these reels should be "mirror" reels. If one changes all the "parallel" reels (in "trad" dances) for "mirror" all of the difficult transitions they involve will disappear.
The other reels to which you refer might be termed reels with corners. If one starts with first corners they are left shoulder, with second, right.