Season turkey wings with salt and pepper.
Heat oil or butter in a large skillet or Dutch oven over medium-high heat and brown the wings on all sides until golden, about 5 to 7 minutes.
Add sliced onion and optional carrots and celery to the pan and cook until softened, about 5 minutes.
Stir in minced garlic and cook for 1 more minute.
Pour in broth and add thyme or poultry seasoning, bringing the mixture to a simmer.
Reduce heat to low, cover, and cook gently for 45 to 60 minutes until meat is tender and nearly falling from the bone.
Remove wings and keep warm, then whisk flour into the simmering liquid and cook until gravy thickens, adjusting seasoning as needed.
Return wings to the gravy or pour gravy over them, garnish with chopped parsley, and serve with preferred sides.