Position racks in the upper and lower thirds of the oven and preheat to 375 degrees . Layer the eggplant in a colander, sprinkling each layer with salt; let stand for 20 minutes. Rinse, then pat dry.
Place the eggplant slices in a large bowl, drizzle with the olive oil and toss. Arrange on 2 parchment-lined baking sheets and place the garlic cloves in a corner of the pan. Roast until the eggplant is golden-brown, about 30 minutes. Let cool, then peel the garlic.
Meanwhile, arrange 12 bread slices on a baking sheet. Brush with the butter, sprinkle with the caraway seeds and season with salt and pepper. Bake until golden, about 15 minutes.
Using a food processor, grind the remaining 2 bread slices into breadcrumbs. Add the garlic, yogurt and all but 12 of the eggplant slices; process until smooth. Pulse in 1/4 cup parsley; season with salt and pepper.
Spread the eggplant puree on the 12 toasts and top each with an eggplant slice and more parsley.