Heat oil in a large Dutch oven over medium-high. Add onion and carrots, and cook, stirring, until softened, about 5 minutes. Add garlic, and cook, stirring, until fragrant, 1 minute. Add sweet potatoes, Yukon Gold potatoes, salt, black pepper, cumin, cinnamon, and cayenne pepper, and cook, stirring, 1 minute. Add stock; bring to a boil. Reduce heat to medium-low, and simmer until potatoes are very tender, about 30 minutes.
Melt butter in a skillet over medium-high. Add pecans, and cook, stirring often, until toasted, about 8 minutes. Transfer to a bowl.
Transfer potato mixture, in batches, to a blender. Remove center piece of blender lid (to allow steam to escape); secure lid on blender, and place a clean towel over opening in lid. Process until smooth. (Or process soup in Dutch oven using an immersion blender.)
Top servings with 2 tablespoons pecans, 1 tablespoon yogurt, and 1/2 tablespoon parsley.
LEFTOVERS: Refrigerate servings of leftover soup, and reheat in the microwave. Serve with extra toasted pecans, yogurt, and parsley.