This paper presents a series of deployable mechanisms based on n-UU (universal joint) Wren parallel mechanism (PM) units, which undergo one degree-of-freedom (DOF) Borel–Bricard motion. First, the PM unit is developed into ortho-planar mechanisms by adopting an R–R joint. The link parameters of the mechanism are optimized to maximize the folding ratio while avoiding interference. Then, the optimized PM units are piled up to construct a novel 1DOF multilayer mechanism that has the largest folding ratio among similar structures in the literature. Moreover, polyhedral deployable mechanisms are obtained by connecting the PM unit using U joint or UU joint. Apart from saving space, the polyhedral mechanisms can transform among different shapes of polyhedrons. Finally, variations of the n-UU PM are investigated with different shapes of platforms, and each mechanism has its unique movement characteristics.