Bring water and rice mix to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and liquid has been absorbed, 20 to 22 minutes. Transfer rice to a bowl to cool.
Bring a pot of water to a boil; cook peas in the boiling water until just cooked, about 2 minutes. Drain and cool.
Mix cooled peas, artichoke hearts, olives, tomatoes, parsley, and green onions into cooled rice mixture.
Blend vinegar, vegetable oil, and sugar together in a blender or shake in a covered bottle until dressing is emulsified and creamy. Drizzle dressing over salad and toss to coat.