In a medium nonstick skillet, cook the chopped onion in 1 tablespoon olive oil over medium-high heat for 5 minutes. Add the shrimp, season with salt and pepper and cook until just opaque, 2 to 3 minutes. Let the shrimp cool slightly, then coarsely chop; toss with half of the lime peel, lime juice and parsley.
Place a tortilla on a work surface and sprinkle with 1/4 cup cheese. Top with 1/4 cup corn, one-quarter of the shrimp mixture and another 1/4 cup cheese; season with salt and pepper and set another tortilla on top. Repeat with the remaining tortillas, cheese, corn and shrimp mixture to make 4 quesadillas.
In a medium skillet, heat 2 tablespoons olive oil over medium-low heat. Add a quesadilla to the pan and cook, turning once, until crisp, about 3 minutes. Repeat with the remaining olive oil and quesadillas. Let cool slightly; cut into wedges.
In a medium bowl, combine the sliced onion and the avocado. Toss with the remaining lime peel, lime juice and parsley; season with salt and pepper. Serve with the quesadillas.
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