Whisk 1/2 cup olive oil, harissa paste, hot pepper relish, minced garlic, lemon juice, and salt in small bowl to blend. (Can be made 1 day ahead. Cover and refrigerate. Bring to room temperature and rewhisk before using.)
Preheat broiler. Rub toast slices on 1 side with peeled garlic clove, then place 1 toast on each of 4 plates.
Heat remaining 1/4 cup olive oil in large ovenproof nonstick skillet over medium-high heat. Break eggs into skillet; sprinkle lightly with salt. Cook until whites are just set but yolks are still runny, about 2 minutes. Remove from heat; sprinkle mozzarella cheese over eggs. Place skillet under broiler and broil just until cheese melts and centers of yolks are still slightly runny, about 2 minutes. Carefully place 1 egg atop each toast slice. Spoon harissa sauce over eggs. Scatter parsley leaves over and serve.
