Preheat a large stock pot or Dutch oven over medium heat. Using a paring knife, make a shallow incision along the length of the sausage. Peel back the casings and discard them.
Add the olive oil to the pot and heat until shimmering. Add the sausage and break it apart into 1" pieces with a wooden spoon. Cook, stirring occasionally, until browned all over, around 8 minutes. Reserve the fat in the pan, transfer the sausage to a bowl and set aside.
Add the celery, carrots and leeks and season with a pinch of salt. Cook, stirring, until the vegetables are soft and translucent, 10-12 minutes. Add the garlic and cook, stirring, 60 seconds. Add the paprika and chile flakes, if using, and cook, stirring, 60 seconds.
Add the stock, potatoes, browned sausage, bay leaves and thyme and stir to combine. Bring the soup to a simmer, cover with a lid, reduce the heat to medium and cook until the potatoes are fork-tender, 12 to 14 minutes.
Add the kale and cook 3-4 minutes or until wilted to desired texture. Taste the soup for seasoning and adjust with salt and pepper as desired. If the soup is too salty or thick, add ½ cup of water at a time until you reach your desired consistency and taste.
Add the parsley and stir to combine. Ladle the soup into individual serving bowls and enjoy!