Heat oil in a large heavy pot over medium-high heat. Add leeks, celery, carrots, salt and pepper. Cook, stirring often, until the vegetables have softened, 8 to 9 minutes.
Add broth, water and Parmesan rind to the vegetables. Increase heat to high; bring to a boil, stirring occasionally. Reduce heat to medium-high; add pasta and cook until al dente, 9 to 10 minutes. Reduce heat to medium-low; stir in turkey, lemon juice, parsley, chives and tarragon. Cook until the turkey is warmed through, about 2 minutes. Remove and discard the Parmesan rind before serving.