Heat the olive oil in a large skillet over medium-high heat. Add the beef and cook, breaking up with a spoon, until no longer pink, 5 to 7 minutes.
Add the green pepper, tomato paste, brown sugar, and salt. Cook, stirring occasionally, about 5 minutes.
Remove from heat and stir in the chopped tomato, red pepper, and parsley. Serve on buns with lettuce and tomato.