All redundant systems are parallel, by definition. If they were placed in series, then the output of one would become the input of the other. Do you see the problem with that? By placing them in parallel, then both can come up with a solution, i.e. left or right and when. These output results are then often compared by yet another computer and a fault posted if they disagree.
You should be able to mechanize it in either fashion, though a parallel setup would be easier, because the damper actuator could be bolted down at one end and it wouldn't move back and forth with the control linkage like a series actuator would. The important thing is to measure the control displacement [i.e. what did the pilot command, if anything] and the aircraft motion [normally with an accelerometer], and then process the difference against an expected value [determined through flight test and analysis]. You don't want the system to wash out a commanded pilot input (unless you're using it as a rate limiting system), but it is entirely possible to measure control displacement and either add or subtract control surface movement by either a parallel or a series actuator.
So, if you get more acceleration in the yaw axis than you should based on the pedal movement, the yaw damper would reduce the control linkage displacement. If the pilot moved the pedals, but the system didn't sense an appropriate amount of yaw acceleration, the system could increase the control linkage displacement. It would work whether it was in parallel or series.
The ones I've seen were parallel systems using walking beam mechanisms.