Heat 2 tablespoons oil in a large cast-iron skillet over medium-high; cook potatoes in an even layer 12-15 minutes, stirring occasionally, until golden brown and crispy.
Push potatoes to sides; cook sausage in center 5-6 minutes, breaking up with a spoon, until browned and cooked through.
Add remaining 1 tablespoon oil; cook onion and pepper 4-5 minutes until softened, then garlic, paprika, salt, and pepper 1 more minute until fragrant.
Make four wells; crack an egg into each, cover, and cook 4-6 minutes until whites set but yolks remain runny, or longer for hard yolks.
Off heat, sprinkle with cheddar and fresh parsley; serve immediately from skillet.