In a medium pot, heat 2 teaspoons oil over medium-high. Add leeks and season with salt and pepper. Cook, stirring frequently, until leeks are soft and lightly browned, about 6 minutes. Add broth and potatoes and bring to a boil. Reduce to a simmer and cook until potatoes are barely tender, about 10 minutes. Add green beans and cook until crisp-tender, about 6 minutes.

Meanwhile, in a food processor, combine parsley leaves and 1 tablespoon oil and process until finely chopped. Remove soup from heat and stir in parsley puree.