Bring a pot of generously salted water to a rapid boil. Add the orzo and cook according to package directions, adding the corn in the last minute of cooking, until al dente, about 8 minutes. Drain and transfer to a medium bowl. Set aside.
Pour the lime juice over the avocado and coat evenly (this will minimize discoloration). Whisk together the olive oil, vinegar, sugar, cumin, and 1/2 teaspoon salt. Drizzle over the orzo and add the avocado with lime juice, tomatoes, and parsley. Mix until combined and serve warm or at room temperature.