You're right there is some confusion. Even Remington isn't consistent: They call the $1,081
Wingmaster Classic Trap's stock a
Monte Carlo, but for $89 more the same stock in the
1100 Classic Trap is called
Competition-style, go figure.
My friend,
BigJimP, a major proponent of the parallel stock, would be better at describing its myriad benefits. Three advantages are: consistent mounting, a better view of the target and reduced felt recoil (kick) to the cheek.