Heat the olive oil in a skillet over medium heat; cook the green onions, carrot, red bell pepper, and parsley in the hot oil until the green onions are softened, about 10 minutes.
Combine the green onion mixture with the crabmeat and black beans in a large mixing bowl. Stir the egg, bread crumbs, and mayonnaise into the mixture until evenly mixed; divide into 12 approximately equal portions and shape into patties. Refrigerate the patties until set, about 1 hour.
Heat the canola oil in a large skillet over medium-high heat; fry the patties in the hot oil until browned and cooked through, about 5 minutes per side.