I tried using a weapon in both hands: a dagger in one and a sword in the other. But I soon realized that I could not parry, since each mouse button is used to swing a weapon. Is there another way to parry when both hands are occupied?
Dual wield in game-land is always a tradeoff: you are doubling your offense in exchange for defense. So it could be seen as a balancing choice on Bethesda's part. Equally, it could have been forced on us by the people who play the game on consoles and their lack of buttons.
IRL, dual-wielding is done for the opposite reasons. You gain off-hand defense, but you'd not normally want to attack with it.
There is a mod that allows parrying (effectively blocking) while dual wielding. There are several options available on how it will handle using 2 keys to do 3 things.