Make the marinade. In a large bowl, whisk together ½ cup BBQ sauce, 2 tablespoons olive oil, 1 tablespoon lemon juice, 1 teaspoon smoked paprika, 1 teaspoon garlic powder, 1 teaspoon onion powder, 1 teaspoon salt, and ½ teaspoon black pepper.
Marinate the chicken. Add the chicken breasts and turn to coat completely. Cover and refrigerate for at least 30 minutes, or up to 4 hours for deeper flavor.
Prepare the BBQ glaze. In a small bowl, whisk the remaining ¼ cup BBQ sauce with 1 tablespoon honey and 1 tablespoon apple cider vinegar. Set aside ⅓ cup of this mixture for serving.
Prepare the vegetables. Toss the zucchini, red bell pepper, and red onion with 1 tablespoon olive oil, ½ teaspoon salt, and ¼ teaspoon black pepper.
Preheat the grill. Heat a gas grill to medium-high, about 400°F to 425°F, or prepare a charcoal grill for direct and indirect heat. Clean and lightly oil the grates.
Grill the chicken. Remove chicken from the marinade and let excess drip off. Grill chicken breasts for 6 to 8 minutes per side, until grill marks form and the internal temperature reaches 165°F (74°C).
Add the BBQ glaze. During the final 3 to 4 minutes, brush the chicken generously with the prepared BBQ glaze. Flip once and grill until the sauce is sticky and lightly caramelized.
Grill the vegetables. While the chicken rests, grill the vegetables for 6 to 10 minutes, turning occasionally, until tender and lightly charred.
Rest and serve. Transfer the chicken to a plate, cover loosely with foil, and rest for 5 minutes. Serve with grilled vegetables, the reserved BBQ sauce, chopped parsley or cilantro, and lemon wedges.