Place rinsed meat chunks in an extended heavy pot. Barely cover with the orange juice and water, add a teaspoon of salt and set over high heat. Once it comes to a boil, bring the heat down to medium and let is simmer for about 40 to 45 minutes, or until most of the liquid has cooked off and the meat is thoroughly cooked, and has rendered most of its fat.
Meanwhile, remove the stems from the chiles, make a slit down their sides and remove their seeds and veins. Place them in a bowl, cover them with boiling hot water, and let them sit and rehydrate for about 15 minutes. Place the chiles and 1 1/2 cups of their soaking liquid in the blender along with the onion, garlic, parsley, oregano, cumin, black pepper, vinegar, and puree until smooth.
Once the meat is ready, place it in a bowl along with any remaining cooking broth. Once it is cool enough to handle, shred it with your hands or using two forks.
In the same pot, heat oil over medium heat. Pour in the chile sause and let it season and simmer for 4 to 5 minutes. Toss in the shredded meat along with any of its remaining cooking broth. Sprinkle 1/4 teaspoon salt and let it cook, stirring often, until the meat has absorbed most of the chile sauce, which will have thickened, seasoned and changed color to a much darker tone. It will take about 20 minutes. Taste for salt and add more if need be.
Serve with warmed flour tortillas on the side. If you wish, spoon chilorio on tortillas and roll them into burritas or burras. They are wonderful with refried beans and Mexican avocado or guacamole on the side as well.