I did some edge-on-edge testing a few years back using a Windless cavalry sabre and a short sword I made out of 5160. I held the sabre in a vise, and braced the tip against a table. I then took full strength swings down onto the sabre. There was much less damage than I had expected, even on the Windless sabre. I put maybe 1/4" deep dings into the sabre, and >1/16" dings into my sword. I touched up my sword -- you can feel the nicks, but you wouldn't notice them to look at it. I can cut a free-hanging rope with the sword, it is sharp.
I've circled once of the dings. This is after resharpening.
Swords are much less delicate than everyone is assuming here. And the testing I did is completely unrealistic. No one would do a full strength hit directly onto a fixed sword. The whole point of parrying is to deflect the blow, not stop it. Even with something like a St. George parry (blocking an straight hit to your head) your parry will have give.
Mechanically, your hand and sword is much stronger when parrying with the edge. Your hand and wrist can apply much more force in line with the edge than with the flat. In addition, European swords have sword guards aligned with the edge. This makes sense if you are parrying with the edge towards the other blade -- the guard can catch anything you miss.
If you parry with the flat, your opponent can slide up your blade and into your arm.