ARMA has been teaching parries/deflections as edge-against-flat and flat-against-edge since we began, and I have NEVER seen or heard of anyone breaking a steel blade by doing it this way since I started in 2000. Even wasters rarely break from such use. Aside from very thick rapier blades, any flat blade decently well made has a springy character to it in that dimension that will absorb and rebound from a strike with usually minimal effects (and as stated above, human hands have some give to them as well). This is especially true if you receive the blow on the lower, strong portion of the blade as nearly all historical sources instruct. Even in sources such as Di Grassi that say "encounter the blow with your edge", I've found that when you apply proper footwork, distance and timing, you still end up with your edge against the opposing flat even when that's not what you're specifically aiming for. Flat parrying is easy to do, it's reliable, and it consistently improves economy of motion. The alternative is this:
http://www.thearma.org/Videos/EdgeBashing1.htm
Just for clarification, when we say "edge on edge," we are referring to two edges meeting at nearly perpendicular ~90 degree angles. This is what causes the kind of damage seen in the video above. If the edges meet at a highly oblique angle (30 degrees or less, for instance), then the forces are different and the edges are not going to dig into each other. None of us expects every single parry to be a perfect edge-against-flat (which is nearly impossible), just that the opposing forces are predominantly aligned in that direction. Some people will still insist on calling that "edge on edge", but they are going by the letter of the law, not the spirit of it as we intend.