Bring water, quinoa, and 1 teaspoon salt to a boil in a small saucepan. Reduce heat; cover, and simmer until tender and water has been absorbed, about 15 minutes. Transfer to a baking sheet, and let cool on a wire rack.
Combine shallot, vinegar, and red-pepper flakes in a small bowl. Gradually whisk in oil until emulsified.
Transfer quinoa to a large bowl, and stir in cucumber, parsley, and vinaigrette. Season with salt.