Mechanical interference and singularities within the reachable workspace often restrict the orientational workspace of parallel robots. Introducing kinematic redundancy can alleviate this limitation. This paper discusses the possibility to produce unlimited rotation of the platform of a tripedal (6+3)-degree-of-freedom kinematically redundant parallel robot. The articulated platform of such a robot has three degrees of mobility. The platforms considered here are planar linkages that contain either revolute or prismatic joints. It is shown that at least two revolute joints are required to produce unlimited rotation with appropriate design and initial configuration, while the platforms with two prismatic joints cannot produce such rotations without crossing a singularity.