Based on my experiences with various kinds of I/O, including copying files over networks, I am strongly inclined to believe that there is considerable latency to take advantage of. (There is considerable latency just copying data around on disk!) So some naive parallelism would be a considerable win.

Try it. Benchmark it with different numbers of copying programs. Set the actual number of copies that you'll use to a bit below what gives you the best throughput (so that you don't be too much of a hog if other people try to use the network).

Of course if you can use rsync rather than a homegrown solution, by all means do!