Pat the chicken pieces dry and season them with salt and pepper. If you want extra depth, heat the oil in a skillet over medium-high and brown the chicken briefly—about 2 minutes per side—then transfer to the slow cooker; otherwise place the raw pieces directly in the cooker.
In a medium bowl whisk together the cream of chicken soup, sour cream, Dijon mustard, garlic powder, and dried parsley until smooth. Taste and add a pinch more salt or pepper if needed.
Scatter the diced onion over the chicken, pour the sauce mixture on top, and give everything a gentle stir so the chicken is coated and the onion is distributed. Cover the slow cooker.
Cook on LOW for about 5 hours (or HIGH for 2 1/2–3 hours) until the chicken is tender and the sauce is bubbling at the edges. If possible, stir once about halfway through to redistribute the sauce.
While the chicken cooks, toss the stuffing mix with the melted butter in a bowl until evenly moistened but not soggy. Hold at room temperature until ready to top the casserole.
About 30 minutes before the end of the cooking time, spread the buttered stuffing in an even layer over the chicken and replace the lid. For a firmer, slightly browned top, remove the lid for the final 10 minutes to let some steam escape.
If you prefer a crisp, golden crust: transfer the contents to an oven-safe dish and broil for 2–4 minutes, watching closely so it doesn’t burn. Otherwise, let the casserole sit uncovered in the slow cooker for 5–10 minutes to set.
Spoon onto plates or into bowls, sprinkle with fresh parsley, and serve warm. The chicken should pull apart easily and the sauce will be creamy—adjust seasoning at the table if needed.