Rinse the quinoa under cold water. Place it in a saucepan with the water and a pinch of salt. Bring to a boil, then reduce heat, cover, and simmer for 15 minutes until the water is absorbed. Fluff with a fork and set aside.
While the quinoa cooks, mix the olive oil, lemon juice, oregano, salt, and pepper in a small bowl. Brush the mixture over both sides of the chicken breasts.
Heat a grill or grill pan over medium-high heat. Cook the chicken for 6 to 7 minutes per side until it reaches an internal temperature of 165 degrees. Remove from heat and let rest for 5 minutes before slicing.
Divide the cooked quinoa among four bowls. Top each with sliced chicken, cherry tomatoes, cucumber, olives, and feta cheese.
Sprinkle the chopped parsley over the bowls just before serving.