Bring water to a boil in a saucepan. Add quinoa and 1/4 teaspoon salt. Reduce heat to medium-low. Simmer, covered, until quinoa is just tender and liquid is absorbed, 12 to 14 minutes. Remove from heat, uncover, and let cool completely.
Stir together cooled quinoa, tomatoes, cucumber, onion, olives, radishes, oil, lemon juice, mint, parsley, pepper, and remaining 1/4 teaspoon salt in a large bowl. Serve immediately or chill up to 2 hours. Sprinkle with almonds (if using) before serving.