Heat oil in Dutch oven over medium-high heat until shimmering. Add onion, sugar, and ½ teaspoon salt and cook until onion is softened, about 5 minutes.
Stir in garlic, paprika, saffron, ginger, and cumin and cook until fragrant, about 30 seconds.
Stir in chickpeas, potatoes, tomatoes and their juice, and zucchini. Bring to simmer and cook, stirring occasionally, until potatoes are tender, 20 to 30 minutes.
Using wooden spoon, mash some of potatoes against side of pot to thicken soup. Off heat, stir in parsley and season with salt and pepper to taste.
Serve with lemon wedges.
