Heat 1 tablespoon oil in saucepan over medium-high heat. Add onion and garlic, 1/4 teaspoon salt, and a pinch of pepper. Cook, stirring, until translucent, about 3 minutes. Stir in quinoa and cook for 1 minute. Add water. Bring to a boil. Reduce to a simmer and cook, covered, until water is absorbed, about 16 minutes. Remove from heat. Let stand 10 minutes; fluff with fork. Cool completely, about 25 minutes.
Bring a large pot of water to a boil. Add 1 tablespoon salt and beans, cooking until bright green, about 4 minutes. Drain and rinse in cold water.
Combine quinoa, green beans, and parsley. Dress with 1/4 cup olive oil, vinegar, 1/2 teaspoon salt, and pepper.