1. Place the chiles in a bowl and submerge in boiling hot water. Soak for at least 20 minutes, up to an hour. Remove the chiles from the water, reserving the liquids. Rinse off the seeds and skin and discard along with the stems. Place the cleaned chiles in a small food processor along with 1/2 cup of roughly chopped cilantro stems, the garlic, and 1/2 cup of soaking liquids. Puree until you get a fine paste, adding more liquid as necessary.
2. Meanwhile, heat 2 tablespoons of olive oil in a large lidded Dutch oven over a medium-high flame. Saute the onion and carrots until soft, about 5 minutes. Add the chili powder, cumin, oregano, and 1 teaspoon of salt. Cook until fragrant, about 2 minutes. You want the spices to get nice and toasted. Carefully pour in the tomatoes and bring to a simmer.
3. Add the sweet potatoes, parsnips, black beans, black chile paste, and remaining soaking liquid. Stir to combine and bring to a simmer, adding one or two cups of water if necessary. You want a soupy consistency, as the chili will thicken as it simmers.
4. Simmer, partly covered for 20 minutes, until the potatoes and parsnips are tender. Season with salt and cayenne pepper to taste. Simmer 5 minutes more.
5. To serve, ladle the chili into bowls and garnish with cilantro leaves, sour cream, and jack cheese.