Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the eggplant and cook, stirring occasionally, until browned and almost tender, 6 to 8 minutes. Remove the eggplant to a large plate. Add 1 more tablespoon olive oil to the skillet. Add the celery and red onion and season with salt and pepper; cook until crisp-tender, 6 to 8 minutes. Add the tomatoes, capers, 1/4 cup vinegar, the honey and 3/4 cup water. Stir in the eggplant, season with salt and pepper and cook until the vegetables are tender and the mixture thickens, 10 to 12 minutes.
Meanwhile, heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Pat the fish dry, season with salt and pepper and add to the skillet. Cook, undisturbed, until golden brown on the bottom, 4 to 5 minutes. Flip the fish and cook until it is just opaque and flakes easily, 2 to 3 more minutes.
Toss the parsley and remaining 2 teaspoons olive oil and 1 teaspoon vinegar in a small bowl; season with salt and pepper. Divide the caponata among shallow bowls and top each serving with a fish fillet. Top with the parsley and serve with the baguette slices.