Preheat oven to 450 degrees F (230 degrees C). Line a baking sheet with aluminum foil.
Spread parsnips and apples out onto prepared baking sheet. Drizzle olive oil over the top and sprinkle with kosher salt. Toss to coat.
Roast in preheated oven until parsnips and apples are tender and caramelized around the edges, about 30 minutes.
Stir apple-parsnip mixture, chicken broth, and potato together in a large pot; bring to a boil, reduce heat to low, and simmer until potatoes are tender, about 30 minutes.
Puree apple mixture with an immersion blender until soup is smooth. Stir cream into soup and season with cayenne pepper and salt; cook until heated through, 2 to 5 minutes.
