Preheat the oven to 425°F.
Place the potatoes on a large rimmed sheet pan and toss with 1½ tablespoons olive oil, ¾ teaspoon kosher salt, ½ teaspoon black pepper, and ½ teaspoon paprika. Spread them out into a single layer.
Roast the potatoes for 15 minutes so they get a head start.
While the potatoes roast, place the sliced sausage, peppers, and onions into a large bowl.
Add the remaining 1½ tablespoons olive oil, remaining ¾ teaspoon kosher salt, remaining ¼ teaspoon black pepper, oregano, garlic powder, onion powder, crushed red pepper flakes, remaining ½ teaspoon paprika, Dijon mustard, and red wine vinegar. Toss everything until evenly coated.
Remove the pan from the oven after the potatoes have roasted for 15 minutes. Add the sausage, peppers, and onions to the pan and mix everything together so the sausage touches the pan surface in several spots.
Return the pan to the oven and roast for 25 to 30 minutes, stirring once halfway through, until the sausage is browned and the peppers and onions are soft with caramelized edges.
If you want extra browning, switch the oven to broil for 2 to 4 minutes at the end, watching closely so nothing burns.
Sprinkle with chopped parsley just before serving if using.