If making ahead of time, store baba ghanoush in an airtight container in the refrigerator, up to one week. Garnish with parsley just before serving.
Preheat oven to broil. Place eggplants on a rimmed baking sheet lined with aluminum foil. Broil until skin is charred, turning as each side blackens, about 12 minutes.
Reduce oven heat to 425 degrees. Continue cooking until flesh is very soft, 12 to 15 minutes. Remove from oven, and let stand until cool enough to handle. Slit open eggplants. Scrape out seeds with spoon (don't worry if some seeds remain), and discard. Slice off tops, and remove skins; discard.
Place flesh in a food processor, and pulse until smooth. Add remaining ingredients except parsley; pulse a few more times to combine. Transfer to a serving bowl, and garnish with parsley. Serve immediately.