Bring a large pot of lightly salted water to a boil. Cook tortellini in the boiling water until tender, 8 to 10 minutes; drain and transfer to a bowl. Toss tortellini with 1 tablespoon artichoke heart liquid; add artichoke hearts.
Whisk mayonnaise, pimento peppers, parsley, basil, Parmesan cheese, Dijon mustard, black pepper, and salt together in a bowl; gently stir mixture into tortellini mixture. Toss to coat. Add more artichoke heart liquid if desired.
