Working, 1 at a time, scrub any grit from shells under running water. Discard open mussels that do not close with a few taps of the finger and any with cracked shells. Put the mussels in a bowl of cold water.
Heat the oil in a large wide pot over medium-high heat and add the chorizo. Using a wooden spoon or potato masher, mash the chorizo into chunks, so it resembles ground beef. Saute until brown, about 8 minutes. Remove chorizo to a bowl. In the same pot, add the onion and garlic and season with salt and pepper, to taste. Saute 3 minutes to sweat. Deglaze the pan with the wine. Pull the mussels from bowl and transfer them and the water to the pot, do not strain, leaving behind any sand and grit that has sunk to the bottom. Stir to coat. Add the chorizo back into the pot and cover. Cook over medium heat until the mussels open, 5 to 10 minutes. Transfer to a large serving bowl, sprinkle with parsley and serve immediately.