Cook onion and sugar in hot oil in a large nonstick skillet over low heat, stirring often, 20 to 25 minutes or until onion is caramel-colored. Stir in 1/4 cup water, vinegar, and 1/4 teaspoon salt. Set mixture aside, and keep warm.
Combine ground beef, tomato paste, parsley, remaining 1/2 teaspoon salt, and pepper; shape into 4 patties.
Grill, covered with grill lid, over medium-high heat (350° to 400°) 4 to 5 minutes on each side or until no longer pink. Serve on buns with caramelized onion and tomato slices.