Rinse the cabbage, cut it into quarters through the core, remove the core, and slice into bite-sized pieces about 1 inch wide. Slice the kielbasa into 1/4-inch rounds and cut each round in half into half-moons. Dice the onion and celery, slice the carrots into 1/4-inch rounds, mince the garlic, and cube the potatoes into 3/4-inch pieces.
Heat the olive oil in a large Dutch oven over medium-high heat until it shimmers. Add the sliced kielbasa in a single layer and cook for 4 to 5 minutes, stirring only once or twice, until golden brown on both sides. Transfer the sausage to a plate with a slotted spoon, leaving the flavorful drippings in the pot, and spoon off any excess fat beyond about 2 tablespoons.
Reduce the heat to medium and add the diced onion, sliced carrots, and diced celery to the drippings. Cook for 6 to 7 minutes, stirring often, until the onion is soft and translucent, scraping up the browned bits from the bottom of the pot with a wooden spoon.
Stir in the minced garlic, dried thyme, and smoked paprika. Cook for 30 to 60 seconds, stirring constantly, until the mixture smells fragrant, being careful not to burn the garlic.
Add the chopped cabbage to the pot in 2 or 3 handfuls, stirring after each addition so it fits. Cook for 4 to 5 minutes, stirring often, until the cabbage starts to wilt and shrink; do not worry if the pot looks overfull at first.
Stir in the cubed potatoes, diced tomatoes with their juices, bay leaf, black pepper, and 1/2 teaspoon of kosher salt. Pour in the chicken broth and return the browned kielbasa to the pot along with any juices left on the plate, then bring the soup to a boil over high heat.
Reduce the heat to medium-low and partially cover the pot with the lid. Simmer gently for 20 to 25 minutes, stirring occasionally, until a potato cube slides easily off a fork and the cabbage looks silky and tender.
Remove the pot from the heat and fish out the bay leaf. Stir in the apple cider vinegar, then taste and add more salt, 1/4 teaspoon at a time, until it tastes right, adding extra black pepper if you like.
Ladle the hot soup into warm bowls and sprinkle each bowl with fresh chopped parsley. Let the soup cool for a minute or two before eating, then serve with crusty bread for dunking.