Place the quinoa in a fine-mesh strainer and rinse under cold water until it runs clear, 1–2 minutes; drain well.
Warm the vegetable oil in a large pot over medium heat until shimmering, about 1 minute. Add the onion, carrot, and celery; cook, stirring, until softened and lightly golden, 6–8 minutes. Stir in the garlic, ground cumin, and dried oregano; cook until fragrant, 30–60 seconds.
Add it to the pot and stir to coat; toast over medium heat until it smells lightly nutty, 2–3 minutes.
Pour in the vegetable broth, add the bay leaf and potatoes, and bring to a boil over high heat. Reduce to a gentle simmer; cover partially and cook until the potatoes are tender and the grains show their visible tails, 15–20 minutes.
Stir in the peas (frozen) and the kosher salt and black pepper; simmer until the peas are bright and tender, 3–5 minutes. Remove the bay leaf.
Turn off the heat and let the soup rest 5 minutes to settle. Ladle into bowls and top with parsley. Serve hot.