Heat oil in a medium saucepan over medium-high until hot. Add leeks, carrots, and celery; cook, stirring occasionally, until crisp-tender, 6 to 8 minutes. Add stock, chicken thighs, and rice; bring to a boil, then reduce heat to low and simmer until vegetables and rice are tender and chicken is cooked through, about 14 minutes. Remove from heat. Transfer chicken to a plate and let cool slightly. Cover soup partially with a lid. When chicken is cool enough to handle, shred into bite-size pieces. Return to soup and cook just until warmed through, about 1 minute. Serve soup with cheese and parsley.