This article presents the position analysis of a new family of 3-limbed 3-dof parallel manipulators, whose output links exhibit a motion of pure translation with respect to the base. Each limb contains four revolute joints and a prismatic pair, which can be mounted anywhere along the kinematic chain or replaced by a fifth revolute one. A univariate polynomial has been found that solves the direct position problem wherever the actuators are placed. The inverse analysis has been carried out in closed-form for all possible locations of the actuated joints. Finally, numerical examples are provided.