Place eggplants directly over a gas burner on medium-high flame. Turn every 2-3 minutes with long tongs until all sides are completely black and the eggplant collapses when pressed. About 12-15 minutes. (Alternatively, broil on highest rack, turning every 5 minutes, for 20-25 minutes.)
Transfer charred eggplants to a bowl, cover tightly, and let steam for 10 minutes. Allow to cool 10 minutes more.
Peel away all charred skin over a colander. Pull flesh apart into strands with two forks. Let drain in colander for 15-20 minutes, pressing gently to expel excess liquid.
Transfer drained flesh to a cutting board and chop roughly - aim for texture, not a smooth paste.
Move chopped flesh to a bowl. Add garlic, lemon juice, and red wine vinegar. Add olive oil in a thin stream while stirring with a fork. Fold in parsley.
Add salt and pepper. Taste and adjust - it should be assertive. Add lemon or salt if needed.
Cover and rest at room temperature for 20 minutes before serving.
Drizzle with olive oil, scatter parsley, add a pinch of smoked paprika if desired. Serve at room temperature with pita bread.