Some argue that one should never parry with the edge because it can damage the sword. Others say one should always parry with the edge because it gives a stronger parry. But what did historical masters really teach?
First of all, let us set aside absolutes. When someone says never, what they generally mean is "never unless you really, really need to and don’t have any other option". So let us rephrase the question as, "Is it preferable to use the edge or the flat?".
The evidence for flat parries is that the swords in museums generally don’t have damage on the edges. The counter is that museums only put on display the best pieces and the ones actually used are kept in the back room. Of course one could respond with, "Those are the swords used by untrained individuals who didn’t know better," but that begs the question:
If it really matters, why didn’t the masters talk about it?
While I haven’t found any real discussion on the matter, I did come across this passage from Ringeck,
When your adversary strikes at you from his right with an Oberhau, counter with a strike from your right shoulder against it. Strike with your long edge and in your strong. When he is soft at the sword then, thrust into his face along his blade.
— Tobler’s translation of Ringeck’s Commentaries, page 22
Unless your opponent is so kind as to abort is attack and present his flat, there is going to be some edge on edge contact. (Of course if your opponent is going to allow you to strike his flat, all the better for you.)
Next we turn Ringeck’s longsword to Capo Ferro’s rapier,
7) OF THE PARRIES
One parries with the true edge, and, although rarely, with the false edge, in a straight line as in an oblique line; now with the point high, now low, now up, now down; depending on whether one is struck with a thrust or a cut, it is with one or the other of the weapons, or with both; taking care that all the parries require an extended arm, and need to be accompanied with the right leg, followed by the left, and when it occurs to parry with dui tempi, during the tempo in which one parries, one will draw the left foot near to the right, and then while striking, will pass forward with the right.
— Swanger and Wilson’s translation of Capo Ferro’s Great Representation of the Art and Use of Fencing, page 23
Capo Ferro doesn’t mention the flat so much as a single time during this discussion. In fact, throughout the entire book the term "flat" only occurs twice, both times referring to a dagger. Capo Ferro isn’t the only one that doesn’t mention the flat of the sword. Chapter 3 of Fabris (Leoni Translation) is titled "Division of the Sword". It too neglects to mention the flat when it enumerates the parts of the sword and their properties.
Here is another passage from Capo Ferro on point,
The falso, then, is designated in two manners, namely dritto and manco; you can avail yourself of the falso dritto in order to hit the enemy’s sword to the outside, that is, toward his right side; and with the falso manco you will hit toward his left side; however it seems to me, if it occurs to you to parry with a falso dritto, I say that it will be far better to turn well your wrist and parry with the true edge for more safety, and the true edge will turn more quickly; but when you will hit the blow with the falso manco, you will be able to strike them with a thrust as well as a cut, taking care that when you parry with the falso, you parry from the middle of your sword up to the point, and when you parry with the true edge, you must parry with the forte, from the middle of your sword down to the hilt; remember that the mandritti and riversi are done with the motion of the elbow, and in such cases when the measure and tempo support it, with the upper part of the arm.
My conclusion at this time is that one parries with the edge against whatever the opponent happens to offer.