The Gough-Stewart platform has been successfully used in a wide variety of fields ranging from medical to automotive applications. This paper proposes a 6-RRRPRR parallel manipulator with orthogonal non-intersecting RR-joint configurations and ball screw actuators without guide mechanisms. A novel methodology is developed to define the dependent RR-joint variables and a numerical algorithm is employed to calculate the joint variables. The parasitic motion caused by the helical motion of the ball screw can be expressed and solved with vector method. The inverse kinematics of this manipulator can be solved. To verify the effectiveness of the proposed approach, simulations are performed with software package ADAMS. A prototype of this manipulator is manufactured. Its resolution, accuracy, and repeatability are measured. It is shown that the presented method is effective for this parallel manipulator.