In a large pot set over medium-high heat, warm the oil. Add the onion and cook until fragrant, about 1 minute. Add the potatoes, followed by the eggplant. Do not stir. Sprinkle with the paprika, pepper, 2 teaspoons salt, and then the chopped parsley. Do not stir. Cover and cook for 5 minutes.
Place the tomatoes on top and sprinkle with 1 teaspoon salt. Do not stir. Cover and cook until the tomatoes begin to soften, about 7 minutes.
Stir all the vegetables together, cover, and cook until slightly softened, 10 to 15 minutes. Give everything a good stir, scraping and incorporating any browned bits on the bottom of the pot.
Cover and cook until the vegetables are tender, the tomatoes have broken down, and the sauce has thickened, 10 to 15 minutes.
Sprinkle the eggplant stew with parsley, if using, and serve immediately.