Bring a large pot of lightly salted water to a boil. Cook quinoa pasta in boiling water for 2 minutes; add broccoli and carrots to the water. Continue cooking pasta and vegetables until pasta is cooked through but firm to the bite, 4 to 7 minutes; drain. Rinse with cold water until cool; drain. Transfer pasta mixture to a large mixing bowl.

Toss artichoke hearts with marinade, Italian salad dressing, grape tomatoes, cucumber, feta cheese, olives, and parsley with the pasta mixture. Cover bowl with plastic wrap and refrigerate 2 hours to overnight.