Rinse quinoa well. Combine with 1.5 cups water in a saucepan and bring to a boil.
Reduce heat, cover, and simmer until tender and water is absorbed, about 15 minutes. Fluff and cool.
Stir ground flaxseed with 3 tbsp water in a small bowl. Let thicken 5 minutes.
Warm 1 tbsp olive oil in a skillet over medium heat. Soften onion 3 minutes; add garlic for 1 minute. Cool slightly.
Squeeze shredded zucchini very dry with a towel to remove excess moisture.
Combine cooled quinoa, mashed chickpeas, carrot, zucchini, onion, garlic, parsley, Parmesan, lemon zest, paprika, cumin, salt, and pepper.
Fold in the flax mixture and panko until the dough holds together. Rest 5 minutes.
Form 8 small patties, about 1/2 inch thick. Press edges to keep them compact.
Heat remaining 2 tbsp olive oil in a large skillet over medium heat.
Pan-fry patties until deeply golden and crisp, 3–4 minutes per side. Season lightly with salt if needed.