This paper proposes a novel approach to providing series elastic actuation for parallel robots. Current series elastic actuators (SEA) are limited to providing single degree-of-freedom (DoF) series elastic actuation. To provide series elastic actuation for each DoF of a parallel robot, an SEA unit must be added to each active joint. In this paper, we introduce a new approach: we recognize a parallel robot as a multi-axes actuator, and we connect the robot platform and the load through a parallel elastic element which has an approximately linear and decoupled wrench-displacement relationship in each dimension. We first introduce the structure of this elastic element and demonstrate its linearity and decoupling properties through a computer model. A constant diagonal stiffness matrix is obtained to relate the six-dimensional wrench and deflection of the elastic element. Then, an integration between this elastic element and a Stewart Platform is achieved in simulation. An experiment is then performed to control the robot platform to follow an input motion of the load while applying zero wrench on the load, which is inspired by a common task performed by rehabilitation robots. Using this approach, we show that the force control of the robot can be transformed into position control while providing necessary compliance between the robot and its load.