It seems like I strain my supraspinatus too often when bench pressing with my standard bar w/ overhand grip; how do bench presses with a swiss bar compare using a grip parallel to my torso? Would that place less strain on my shoulders?
It would depend on the grip and the width of that grip on the swiss bar.
The suprispinatus abducts the upper arm (Moves it away from the torso), so if you do a swiss bar with a wide grip, you are still going to need to move the upper arm out to accommodate. The primary thing the swiss bar would achieve is rotation of the grip (And thus the forearm), which would not really have a significant impact on the overall motion.
There are alternatives (with better results) than the traditional bench press, such as flyes, crossovers, things of that nature. However, anytime the upper arm moves away, you're involving the supra. Since it's primarily a stabilizer, that you strain it quite often indicates to me that either you're using too much weight or you have a form break somewhere. I'd recommend having someone take a look at your form for evaluation.