An axisymmetric vectoring exhaust nozzle (AVEN) driven by a 3-SPS/3-PRS parallel manipulator is designed to universally vector the exhaust flow of a variable-geometry Laval nozzle (VGLN) surrounded by multiple groups of convergent and divergent flaps suspended by RSRR kinematic chains. The inherent singularities in parallel manipulators cause the 3-SPS/3-PRS parallel manipulator and the RSRR kinematic chains to produce configuration bifurcations, deeply hidden in parallel manipulators and difficult to perceive. They cause the mobile platform not to be controlled by the input parameters at the bifurcation points, leading to disastrous consequences to the flight safety and control reliability of the parallel manipulator applied in the aerospace field. To explore the configuration bifurcation characteristics concealed in the VGLN and RSRR kinematic chains, the joint coordinate variables of the RSRR kinematic chains and the output parameters of the VGLN are employed as bifurcation variables. The configuration bifurcation points corresponding to these bifurcation variables have been marked on their respective configuration curves using the extended equation and the homotopy method. This study reveals, for the first time, the adverse effects of the bifurcation behaviors associated with the RSRR kinematic chains and the VGLN on the gained singular movements and the flight safety of jets equipped with the AVEN. By analyzing the generating mechanisms of singular movements at the kinematic joints in the bifurcation points, several constructive suggestions are proposed to enhance the flight safety and control reliability of the AVEN.