Place eggs in a saucepan and cover with cold water. Bring to a gentle boil, then cook for 10 minutes. Transfer immediately to cold water and let cool.
Peel the eggs and chop them into small pieces. Place in a mixing bowl.
Add mayonnaise, Dijon mustard, lemon juice, salt, and black pepper. Mix gently to combine.
Stir in chopped celery, red onion, and parsley. Mix carefully without mashing the eggs too much.
Taste and adjust seasoning if needed. Chill for 15 to 20 minutes for best flavor.
Place lettuce on bread if using, add egg salad, spread evenly, and top with another slice of bread.
Slice and serve immediately.