Bring a large pot of lightly salted water to a boil. Add pasta and cook for 8 to 10 minutes or until al dente; drain.
Meanwhile, heat oil in a small skillet over medium-low heat. Stir in sesame seeds and cook until golden brown. Remove from heat. Stir in soy sauce, vinegar, and sugar. Pour dressing into a sealable container, and set aside.
In a large bowl, mix together pasta, cooked chicken, and 1 cup dressing (reserve remaining dressing). Cover salad, and refrigerate at least 6 hours.
Directly before serving, stir in parsley, green onions, and spinach. Toss with remaining dressing, if desired.