The advantages are a thinner board for any given volume which gives a lower centre of gravity ( greatly improving control and feel ) as well as improving flex.
Greater flex occurs when thickness is evenly distributed, and the flex pattern is much better than when flex is confined to a small area in the tail as occurs with tapered profile boards.
With a parallel profile board the rider's feet are always at a constant distance from the bottom of the board, rather than standing facing uphill or downhill in relation to the bottom as with tapered profile boarads. The surface one is standing on is the same shape as the surface which is contacting the water. . . this makes for more intuitive surfing.
The constant rail sections which I use are not required for the benefits of the parallel profile, but do work naturally with the profile.