Accurate and efficient kinematic solutions are essential for stable real-time control in robotics. While inverse kinematics (IK) in parallel robots is straightforward, forward kinematics (FK) remains challenging due to parasitic motions and passive joint dependencies. This paper addresses the FK problem of the 3-RPS parallel robot, where existing methods (algebraic, Newton-based numerical, and iterative approaches) face distinct challenges: algebraic methods suffer from multiple solutions, while Newton-based and iterative approaches are hindered by dependence on initial values and high computational demand. We present the Efficient Iterative Method for FK (EIM-FK) for the 3-RPS parallel robot, which requires 40% fewer arithmetic operations per iteration compared to traditional methods. Additionally, our proposed novel initial value estimation enhances the performance of the standard Newton-based method. The artificial neural network (ANN) is also applied to obtain the FK solution to expand the computational alternatives. Simulations compare EIM-FK, the improved conventional method, and the ANN approach in terms of accuracy and computational efficiency. Although both the improved conventional method and the ANN can solve FK, EIM-FK achieves superior accuracy with significantly lower complexity, requiring only three iterations on average to converge within an accuracy of 10⁻⁶. Thus, our method outperforms existing solutions in accuracy and computational efficiency, positioning it as a superior solution for real-time control.