Place diced avocado in a medium bowl, Mash thoroughly with lime juice, and salt and pepper to taste. Divide smashed avocado between toast slices. Top toast with tomato and pepper slices. Set aside for a moment.
Make the fried eggs: Working in batches, heat oil in a medium saucepan over medium-high heat until shimmery. Crack 2 eggs into pan and sprinkle eggs with salt and pepper. Allow eggs to cook, without flipping until whites of eggs are set. Carefully remove from pan with a spatula and transfer to a plate. Repeat step 2 with remaining 2 eggs.
Carefully arrange fried eggs on top of each toast. Sprinkle minced parsley over top and enjoy toast immediately!