In a blender or food processor, combine soy sauce, ginger, honey, sesame oil, garlic clove, and black pepper. Process until smooth. Pour mixture into a plastic bag, add flank steak and shake to coat. Refrigerate 15 minutes and up to 24 hours.
To make the mojo: In a blender or food processor, combine cilantro, mint, green onions, water, lime juice, olive oil, and garlic clove. Process until smooth.
Coat an outdoor grill or stovetop grill pan with cooking spray and preheat to medium-high. Remove steak from marinade (discard marinade) and grill 5 minutes per side for medium meat. Arrange carrots and parsnips alongside steak and grill for 10 minutes, until crisp-tender, turning frequently.
Let steak stand 10 minutes before slicing crosswise into 1/4-inch thick slices.
Serve steak with mojo drizzled over top and sliced carrots and parsnips on the side.