Wash and then dice the potatoes into bite-sized pieces. Place the diced potatoes into a large saucepan or pot.
Fill the saucepan with enough water to cover the potatoes and bring to a boil. Cook until tender, about 12 to 14 minutes.
While the potatoes are cooking, prepare the herbed vinaigrette: Place the parsley, basil, green onions, oil, vinegar and salt and pepper into a small canning jar. Place the lid tightly on the jar and shake well. Set aside.
Drain the potatoes and place them into a large bowl. To the bowl, add the artichoke hearts, peas and arugula.
Give the canning jar one more good shake, then pour the vinaigrette over the potatoes and veggies and stir well.
Sprinkle the shaved Parmesan cheese on top and add a bit more salt and pepper if you'd like, then gently toss everything together.