Cook onions and sugar in hot oil over low heat 20 to 25 minutes until onion is caramelized, stirring often. Stir in the 1/4 cup water, vinegar, and 1/4 teaspoon of the salt. Set aside and keep warm.
Combine beef, parsley, tomato paste, Worcestershire sauce, 1/2 teaspoon salt, and pepper. Shape into 4 patties.
Grill over medium-hot heat until beef is no longer pink.
Serve on buns with caramelized onions, tomato slices and lettuce.