Here's a new take on the dual weapons discussion: Defensive pole-weapons.
The ultimate defensive weapon is, of course, the
madu, but you can reach a similar effect by holding a spear, polearm, or even great-sword in the off hand- known as "fighting Christian", because the upside-down great-sword looks like a cross.
The way it works is that you hold the weapon point-down about a third or a quarter from the top, unless it's a mass-weapon like an pole-axe or hammer, in which case you hold it right-side up. For a sword, you hold it on the blade, so that the cross-guard is resting on top of your hand- medieval blades weren't all that sharp, and certainly not just above the hilt, so there's no worry about cutting your hand. Great-swords even have handy leather strapping there so you can choke-up for close-quarter combat.
The bonus for this style of combat is that your "shield" is much faster than the usual board shield. The down-side is that unless you're fighting Christian with those handy cross-guards, you have very little defense against overhead swings and you have to use your regular weapon for that. The secondary bonus is that you can do a damaging strike with your off-hand weapon. For thrusting weapons, you use a "Psycho"-esque stabbing motion and with bladed weapons like axes, you just lean the blade of the weapon with all your weight into the opponent's head.
I know a guy that fights (in the SCA) with two single-hand axes and he holds the left-hand one just below the blade and used the haft for blocking. Both have spikes at the top for thrusting. Both have butt-spikes. That's eight striking surfaces.
Oh, and he hits hard. Ouch.
Btw, the idea of parrying is definitely period, and it takes VERY little force to move the direction of an overhand chop enough so that it misses your head, and yes, that leaves your opponent EXTREMELY vulnerable. There's a reason you don't do that with mass weapons. With pole-weapons and swords it's a little different because of that beautiful thing called an "Oar-stroke" (put the blade of your weapon behind your opponent's head, walk past him, and back-hand him in the back of his head like you're rowing a boat), but for heavier axes and hammers, over-hand swings should
always mean someone dies- either the sap that wasn't fast enough to block it or the sap that completely overextended himself and is now bent over into the oncoming couner-attack.