In a blender or food processor, combine the avocado, sour cream, lime juice, garlic, parsley, and salt, and pulse until smooth. The consistency should be similar to cake batter--thick yet spreadable. To thin it if necessary, add a tablespoon or two of water and pulse to incorporate. Cover and set aside until ready to serve. The crema can be made up to 1 day in advance if covered and stored in an airtight container in the refrigerator.
In a small bowl, combine the brown sugar, chili powder, salt, and the cayenne pepper. Lay the salmon on a work surface and sprinkle the brown sugar mixture evenly over each filet, rubbing it into the flesh on all sides.
In a large skillet, heat 1 tablespoon of the oil over medium high heat. Add 2 of the salmon filets (skin-side-up if they have skin on) and cook, undisturbed, until the underside is crisp and just beginning to blacken, about 4 minutes. Flip the filets and cook an additional 4 minutes, until the fish feels firm to the touch. Remove to a plate immediately. Add the remaining tablespoon of oil to the pan and swirl to coat. Add the remaining 2 filets of salmon and repeat the cooking process. Once all of the salmon is cooked and transferred to a plate, add the lime slices to the now-empty pan and cook just until they begin to caramelize, about 30 seconds. Flip and cook for an additional 30 seconds.
To serve, divide the salmon filets among four plates and spoon 2 generous tablespoons of crema over each. Press a caramelized lime slice into the crema to garnish.