Place eggs in a saucepan and cover with water. Bring to a boil over high heat, and cook 1 minute. Cover, turn off heat, and let stand for 9 to 10 minutes. Drain and add eggs to a bowl of ice water. Peel eggs and set aside.
In a large Dutch oven, add sausage and cook over medium-high heat until fat has rendered and sausage is browned. Remove sausage with a slotted spoon to a paper towel-lined plate and set aside.
Melt butter in the same pot. Add onions and cook, stirring occasionally, until onion has softened, about 3 minutes. Add garlic and cook 1 minute longer. Add Old Bay, 2 1/2 teaspoons Cajun seasoning, 1 teaspoon paprika, oregano, and 1 1/2 teaspoons lemon pepper and stir to combine. Cook until fragrant, about 1 minute. Add stock, water, and potatoes. Add more water if necessary, enough just to cover potatoes. Stir to combine and bring mixture to a boil. Reduce heat slightly to maintain a simmer and cook for 10 minutes. Add corn and continue to cook 10 minutes. Add in reserved eggs and sausage and let simmer gently while wings are prepared.
Preheat an air fryer to 360 degrees F (182 degrees C).
Toss wings with oil and remaining Cajun seasoning, paprika, lemon pepper, salt and pepper. Air fry wings cooked through, crispy, and golden brown, about 15 minutes.
Use a spider or strainer to transfer onions, sausage, potatoes, corn, and eggs from pot to a large, deep serving platter. Add wings to platter and pour over cooking liquid remaining in the pot. Sprinkle with chopped parsley.