I've noticed that in a lot of bikes (though certainly not all), the seat stay and the down tube are parallel or nearly so. Is there any structural reason why a manufacturer might make that choice or is it a more or less arbitrary choice?
The two are not directly related. On a conventional diamond frame the downtube angle is dependent on the length of the top tube and seat tube, and, of course, the seat tube angle. The length of the seat stay is dependent on the seat tube length and angle, and the length of the chain stay.
Obviously, the two "share" the seat tube length and angle (which is related in part to the overall bike "size"), but the length of the top tube is determined by the desired "reach" of the bike, while the length of the chain stay is determined by the desired overall wheelbase (longer for road bikes, eg), and the wheel size and desired clearance (eg, for fenders) between wheel and seat tube.
If you look at Google images for bikes you will see a wide variation in relative angles. Usually the seat stay is at a steeper angle than the downtube.
While I'm not a frame builder or any type of engineer, I can point a couple reasons:
Geometry: The seat stay angle is directly related to the length of the chain stays, which play an important role in the "feel" of a bike. Also the chain stay length changes proportionately with the down tube as the bike size changes; therefore, all frame sizes will retain roughly the same shape.
Aesthetics: Parallel lines are held to be more visually appealing. Look an any example of classic architecture or engineering and you'll see the lines.
I'm sure there's also some structural reasons about balancing forces and stresses on nthe bike (which probably explains why the seat stays usually meet the seat tube around the same spot as the top tube).
Yes, I'd say there are structural reasons for those lines being roughly parallel and it is not arbitrary. (Though the usefulness of my answer depends a bit on my interpretation of your phrase "nearly so.")
I think the similar angles of the tubes you mention has most to do with the strength of triangles and the design of the classic and ubiquitous "diamond" bike frame design which utilizes those triangles such that they accommodate the wheel size that is currently popular in a structurally sound configuration.
The diamond frame also has some economy of materials - aiming to draw straight lines from point to point, creating those triangles efficiently.
There is novelty and utility in other designs (step-through frames, folding bikes) but there are trade-offs and considerations that prevents the geometry choices from being arbitrary.