Toss eggplant with salt and place in a colander set over a bowl. Let sit, about 30 minutes. Rinse and pat dry.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add celery; cook, stirring often, until softened, about 4 minutes. Add onion and garlic; cook and stir until onion is soft and lightly golden, about 5 minutes. Transfer mixture to a bowl using a slotted spoon.
Heat remaining 2 tablespoons olive oil in the skillet. Add eggplant and cook, stirring constantly, until lightly browned, 5 to 7 minutes. Stir in celery mixture, tomatoes, olives, capers, tomato paste, and oregano. Bring to a boil; reduce heat to low and simmer uncovered until caponata is thickened, about 15 minutes.
Season caponata with vinegar, sugar, salt, and black pepper. Transfer to a serving bowl and garnish with parsley.
