Variable Stiffness Actuators prove invaluable for robotics applications in unstructured environments, fostering safe interactions and enhancing task adaptability. Nevertheless, their mechanical design inevitably results in larger and heavier structures compared to classical rigid actuators. This paper introduces a novel 3 Degrees of Freedom (DoFs) parallel wrist that achieves variable stiffness through redundant elastic actuation. Leveraging its parallel architecture, the device employs only four motors, rendering it compact and lightweight. This characteristic makes it particularly well-suited for applications in prosthetics or humanoid robotics. The manuscript delves into the theoretical model of the device and proposes a sophisticated control strategy for independent regulation of joint position and stiffness. Furthermore, it validates the proposed controller through simulation, utilizing a comprehensive analysis of the system dynamics. The reported results affirm the ability of the device to achieve high accuracy and disturbance rejection in rigid configurations while minimizing interaction forces with its compliant behavior.