Different to currently existing serial manipulators, the actuators for some parallel manipulators can be assembled directly for base pairs or prismatic pairs near the base platform. Such a base-mounted actuator or near-base actuator structure will reduce the extra load, thinner links and promote the speed of the device, which will, consequently, promote the production efficiency for a manipulator. Recently, researchers have paid more and more attention to lower-mobility parallel manipulators. However, 5-DoF 3R2T (three rotational, two translational) parallel manipulators with fully symmetrical structures have not yet received enough attention, although they have a wide application background. This study identified 11 5-DoF 3R2T parallel manipulators from a group of about 70 existing structures and proposed seven novel ones. Fifteen of them adopt a base-mounted actuator structure. The other three adopt a near-base actuator structure. All of them have fully symmetrical structures. Moreover, a simple method for all 18 manipulators to avoid assembly singularity is also illustrated in this study.