Prepare the Crab Cake Filling: In a large mixing bowl, combine the crabmeat, breadcrumbs, mayonnaise, egg, Dijon mustard, Worcestershire sauce, lemon juice, Old Bay seasoning, salt, pepper, green onions, red bell pepper, and parsley. Mix gently until well combined.
Assemble the Egg Rolls: Lay an egg roll wrapper on a clean surface with one corner pointing towards you (diamond shape). Place about 2 tablespoons of the crab cake filling in the center of the wrapper. Fold the bottom corner up over the filling, then fold in the sides. Brush the top corner with the beaten egg wash and roll tightly to seal. Repeat with the remaining wrappers and filling.
Fry the Egg Rolls: Heat about 1-2 inches of vegetable oil in a large skillet or deep fryer to 350°F (175°C). Fry the egg rolls in batches, turning occasionally, until golden brown and crispy, about 3-4 minutes per batch. Use a slotted spoon to transfer the egg rolls to a baking sheet lined with paper towels to drain.
Make the Dipping Sauce: In a small bowl, whisk together the mayonnaise, Dijon mustard, lemon juice, sweet pickle relish, hot sauce (if using), salt, and pepper. Adjust the seasoning to taste.
Serve: Serve the crab cake egg rolls hot with the dipping sauce on the side.