The pleasantly sweet-and-sour pomegranate molasses in the tangy vinaigrette here is made by reducing pomegranate juice into a syrup. Look for it at well-stocked grocery stores and Middle Eastern markets.
Bring water to a boil in a medium saucepan over high heat. Stir in bulgur and 1/2 teaspoon salt. Cover, reduce heat to maintain a simmer and cook until the liquid is absorbed, 12 to 15 minutes. Remove from heat and let stand, covered, for 5 minutes. Fluff with a fork and let cool for 5 minutes.
Meanwhile, whisk lemon juice, oil, pomegranate molasses (or balsamic glaze), pepper and the remaining 1/4 teaspoon salt in a large bowl. Stir in shallot. Add the bulgur, tomatoes, beans, cucumber, 3/4 cup parsley and 1/4 cup mint and stir to combine. Let stand at room temperature for 1 hour or refrigerate for up to 1 day.
Serve at room temperature topped with the remaining 1/4 cup each mint and parsley.
Refrigerate for up to 1 day.