Pat the thighs dry and season with salt and pepper. Heat a large, deep skillet over medium, add 1 tbsp. of the EVOO and swirl to coat. Add the thighs, skin side down, and cook, undisturbed, 8 to 10 minutes; transfer to a plate.
Add the eggplant to the pan and cook until soft, about 5 minutes; transfer to a bowl. Add the remaining 1 tbsp. EVOO, the bell pepper and onion; cook, stirring, 2 minutes. Add the zucchini, season, and cook, stirring occasionally, until zucchini begins to soften, about 5 minutes. Stir in the tomatoes and reserved eggplant, then nestle the thighs, skin side up, in the vegetable mixture. Cover and bring to a brisk simmer. Reduce heat to low and simmer gently, partially covered, until the vegetables are very soft and the thigh meat pulls apart easily from the bone, about 45 minutes.
Transfer the thighs to a bowl; remove and discard skin and bones. Using two forks, pull the meat into thick shreds. Return the meat to the pan and stir in the parsley; season. Serve with the grated cheese.