Preheat the oven to 350 degrees F. Line a baking sheet with parchment or aluminum foil.
Place the eggplant on the baking sheet and rub all over with 1 teaspoon of the oil. Rub 1 teaspoon of the oil onto the cut top of the garlic, sprinkle lightly with salt and pepper and place cut side down on the baking sheet. Bake until the eggplant is very soft and turns a purplish-brown color, turning occasionally, and the garlic cloves are soft and golden brown, about 45 minutes. Remove from the oven and let sit until cool enough to handle.
Squeeze each head of garlic into a bowl and discard the papery skins. Cut the eggplant in half lengthwise. Roughly chop the flesh and discard the skin. In a blender or food processor, combine the eggplant flesh and garlic. Add 2 tablespoons of the extra-virgin olive oil, 1 tablespoon of the lemon juice, and parsley and process on high until smooth, adding more oil as desired for a smoother consistency and more lemon juice for flavor. Setaside.