Heat oven to 400°F. Rub chicken with oil, 1/2 teaspoon salt, and pepper; place skin-side up in a small roasting pan. Roast 35 to 45 minutes or until thickest breast meat reaches 165°F; rest 10 minutes.
Meanwhile, boil rice water in a saucepan. Add wild rice; cover and gently simmer 45 to 55 minutes until tender with many split grains. Drain remaining water.
Melt butter in a soup pot over medium. Cook leeks 7 minutes; add parsnips, broth, thyme, and remaining salt. Simmer partly covered 20 to 25 minutes until parsnips soften.
Discard chicken skin and bones. Shred meat into short pieces; retain clear roasting juices.
Blend parsnip-leek broth smooth with an immersion blender. Fold in cooked rice, chicken, and roasting juices.
Add cream; heat gently 5 minutes until steaming throughout without a hard boil. Divide among 6 bowls.