Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with aluminum foil and brush with olive oil.
Brush eggplant slices on both sides with olive oil and place on the baking tray in a single layer.
Bake in the preheated oven until soft and browned, about 20 minutes, turning slices over once after 10 minutes. Transfer slices to a large serving place, brush with vinegar, and sprinkle with salt. Let cool.
Heat a drizzle of olive oil in a skillet over medium-low heat and cook garlic until softened and fragrant without browning, about 3 minutes.
Mix parsley and walnuts in a bowl. Add sauteed garlic and a little bit of olive oil to make a paste. Mix well, season with salt, and spread the paste onto the cooled eggplant slices.