I found at least one other person trying to find a way to extract a JPG snapshot out of a flash video using ffmpeg.

The above article suggests using ffmpeg to extract a PNG, and then convert that to a JPG (since the JPG will be much smaller). Instead, just use the correct arguments to ffmpeg:

ffmpeg -i movie.flv-vcodec mjpeg -vframes 1 -an -f rawvideo -s 320x240 movie.jpg

With -vcodec mjpeg the important argument.