How can I assemble 2 opposing gears so that the axle rotations are perfectly parallel?
How can I get 2 opposing gear's axles to be perfectly parallel? Are there any good workarounds?


You could use an idler gear. It will even out the offset. 

Yet, at the same time, it (or any odd number of idler gears) will cause both axles to spin in one direction. 



On the other hand, two (or any even number of) idler gears will not affect the asynchrony of the axles, but will bring back the offset. 

Then again, placing a single idler gear crosswise will compensate the offset without changing the desired clockwise/ counterclockwise movement. 



Besides, here's two 8 tooth idler gears in line. Both axles seem to be perfectly in parallel to each other. 



Unfortunately, that's actually because of the large amount of play involved. So this solution just benefits from the wiggle and does not run too reliably. 

Edit:


  Are there any older or special gears that are useful here? 


At least theoretically. 



Having an uneven number of teeth, these nowadays obsolete gears from the early 70s would work as well.

As you can tell from the overlapping grey gears, the axles are quite parallel (minus some backlash).