The Paralever suspension was introduced on the 1988 R80GS and R100GS, then used on the 4-valve K and Oilhead bikes (except the R1200C). Whereas the original Monolever system had a single pivot at the front of the swingarm, like most motorcycles, the Paralever uses two links to connect the rear drive to the transmission. This is a "Four-bar" linkage, similar to double-wishbone suspension in cars which keeps the rear drive at a constant angle as the suspension moves. In fact, there are several similarities between the two that can be explored.
Let's be clear on some terms I'll use here. The "rear drive" is the bevel-gear containing casting that is at the end of the swingarm; it contains the flange that the wheel bolts up to. Whenever I refer to "rotation", I am specifically referring to the rotation, or angular change in position, of the rear drive unit as the suspension moves up and down. The wheel rotates while you ride but that's not important, and the Paralever links rotate around their end points also, but don't let that distract you. We are only concerned with the change in the angle of the rear drive unit (shown as a circle in the wireframe diagrams).
Also, I will mention the forces acting on the rear drive: there is a road force (pushes forward on the tire's contact patch during acceleration) and this is balanced by the inertial force of the bike which pushes backwards at the swingarm pivot. These two forces, since they are not directly opposing each other, create a twisting force or torque in the swingarm/drive/wheel assembly that we're looking at.
Lastly, it's worth keeping in mind that most of my diagrams portray the two Paralever links as being parallel and of equal length. This isn't exactly correct, but making this assumption clarifies the difference. At the end I'll show how the actual geometry differs.
Refer to the diagrams below while we look at the basic concept: with the Monolever design, you can see that as the rear drive moves vertically, it rotates.
This is a simple result of the fact that the swingarm and rear drive are one unit, hinged at one end: vertical movement will cause rotation, and rotation will cause vertical movement. Think harder about the latter part of that relationship: the torque of the wheel and drive will move the suspension vertically. This torque-induced suspension movement, sometimes called "the shaft effect" or "shaft jacking" is considered to be a drawback to traditional shaft-drive designs. The Paralever drive unit, on the other hand, does not rotate as the suspension moves up and down - it stays at the same angle as the chassis because of the parallel links. Since there is no direct relationship between vertical suspension movement and drive unit rotation, the torque of the wheel does NOT result in suspension movement. This, in a nutshell, is what the Paralever is about. Understand this paragraph and you're almost finished.
If that didn't work, try this: Think about the motion of the contact patch with respect to the bike. Since the Monolever is one solid piece (in our example), every part of that system rotates around the swingarm pivot. As the contact patch pushes forward, it can actually move forward with respect to the rest of the motorcycle by pushing downwards.
But the Paralever isn't a solid system; the drive is moving without rotating, and each part is moving in the same arc. In fact, the contact patch and rear drive are moving around points that are in thin air! Looking at the arc of motion of the contact patch, you can see that there is no real way for the wheel to try to "scoot underneath" the bike, as the Monolever does. In fact, this "virtual pivot point" is really at the heart of how the Paralever works; BMW has enabled the rear end to pivot around a location where they simply could not put a mechanical pivot.
Here's a automobile example of the same concept. The Monolever is like the "swingarm" rear suspension used on the Corvair. Remember "Unsafe at Any Speed"? The cornering forces tended to raise the rear of that car in exactly the same way, leading to the rollover issue.
And the Paralever is just like the double A-arm suspension, which is one way the auto makers solved the handling problems inherent to the swingarm. Interestingly enough, another solution to the problem is simply to make very long swingarms, like GM used on the front of their trucks for many years. The torque generated by the cornering force is resisted by a longer lever, resulting in less force at the end of that lever (and less lifting). And the parallel to motorcycles? The R1200C never got the Paralever because BMW considered it unnecessary with the longer Monolever swingarm it uses.
Real World functionality
Now to be truly accurate, and get to the real nitty-gritty of the design: the geometry of the Paralever isn't an exact parallelogram. As shown below and in the original picture, the forward pivots are closer together, somewhat negating the theoretical benefit. Consider the two extremes: if the two forward pivots were in the same place, forming a triangle out of the rear drive, swingarm and torque arm, all three of those components would rotate around that point exactly like a Monolever. If the linkages form a true paralellogram, like some of my illustrations show, the rear drive won't rotate at all, which would be like having a swingarm of infinite length. By moving the forward pivots closer together (going partway back to being a Monolever) they create the effect of having a much longer swingarm than they really have. The "virtual pivot point" of this system is somewhere around the steering head - a very long swingarm indeed, which resists the shaft effect as mentioned. Additionally, moving the pivot points closer together changes a few things. For one, they gain ground clearance: with the true parallelogram arrangement I described earlier, the forward pivot would just be too low when the suspension compressed over a bump. Primarily, though, they reintroduce enough "shaft effect" to counteract the squat that would otherwise happen under acceleration.
As shown here, the amount of rear drive rotation is cut roughly in half with the Paralever. This means that torque-induced rotation is smaller and will play a smaller role in inducing vertical suspension movement, but there is still enough to counteract the squat that would otherwise happen under acceleration, as weight is transfered to the rear wheel.
Some people worry about the two U-joints not being at the same angles, and are convinced that this is a mistake on BMW's part.
The two U-joints in the BMW shaft were never intended to be at the same angle all of the time. Yes, theoretically perfect U-joint cancellation only happens then, but it doesn't happen on our bikes because the suspension designers didn't want it.
The intent of the Paralever is to create a longer effective swingarm (with this length determining the amount of rotation that the final drive experiences as the suspension moves). Making a perfect parallelogram creates a swingarm of infinite length, as far as any shaft-jacking effects go, with no rotation of the rear drive at all. We don't want that because we want a certain amount of jacking to offset the weight changes when we accelerate and decelerate, and we need that rotation to provide it. Putting the effective swingarm pivot up by the center of mass of the bike does this the best.
Rotation of the R100GS Paralever is about half what it is with the K75 Monolever.
So yes, the design is not perfect. But keep in mind that the monolever designs use only one U-joint and therefore have much worse driveline velocity fluctuation. Yes they have rubber dampers, but so does the BMW R100GS and I think the R1100GS, and both of those bikes will probably (haven't done all of the math yet) have less fluctuation than the Monolevers.
This is a separate issue from the ends not being in alignment... as in, not being phased correctly.