How does parrying work for a Dual-wielding character in Skyrim?

Parrying or blocking doesn't work when dual wielding.

Source

You cannot block if you have a weapon or spell equipped in your left hand.

There's almost nothing that is exclusive to dual wielding. In the one-handed combat perk tree, there are only two perks that have anything to do with dual wielding.

There's nothing stopping you from just picking up a shield and start blocking.