I was double-clicking on one of the actual PAR2 files with content, rather than the base -- but I think I was doing it directly in Agent (which usually works). I've gone out and come back in the meantime, so I'm not sure that's the only thing I did or not.
I guess it must have been, because now when I double-click on the PAR2 file with 23 blocks, it verifies and works fine. Thanks, greencapt!
PAR2 files are pretty much magic, but sometimes I feel like I gotta go back and write my OWLs again...