Place zucchini in a colander or fine-mesh sieve set over a bowl; season with 1/2 tsp. salt. Let sit 15 to 30 minutes. Rinse off excess salt, then thoroughly pat dry with paper towels.
Meanwhile, prepare a grill for medium-high heat; preheat 5 minutes.
In a small saucepan over medium heat, heat 2 Tbsp. oil. Cook shallot and garlic, stirring occasionally, until light golden and fragrant, about 2 minutes. Add whole-grain mustard, honey, Dijon mustard, paprika, and red pepper flakes and stir to combine. Bring to a boil, then continue to boil over medium heat, stirring frequently, until thickened and slightly darkened in color, about 2 minutes. Let cool slightly.
In a medium bowl, toss zucchini, tomatoes, 1/2 tsp. salt, 1/4 tsp. black pepper, and remaining 2 Tbsp. oil. Season chicken with 1 1/2 tsp. salt and 1/2 tsp. black pepper.
Arrange 4 (16"-x-12") sheets of foil on a flat surface. Pull up sides to form a shallow boat. Divide zucchini mixture among packets. Nestle 1 chicken breast next to vegetables. Spread tops of chicken with honey mustard sauce. Tightly fold foil packets to seal.
Grill, covered and rotating packets halfway through, until an instant-read thermometer inserted into thickest part of chicken registers 165º, 18 to 20 minutes. Carefully open foil packets and sprinkle with parsley.