A 3-RRPRR variable spherical symmetrical parallel mechanism (PM) with arc-shaped sliding pairs and no parasitic motion is presented, exhibiting two rotational and one translational (2R1T) degrees of freedom. Three limbs are symmetrically distributed between the base and end-effector; upper and lower parts of each limb are mirror symmetrical around the middle. The geometry, mobility, forward/inverse kinematics, workspace, and parasitic motion of the mechanism are analyzed, showing its ability to achieve large rotations around a continuous rotation axis. Finally, a structure synthesis strategy for variable spherical symmetrical PM is proposed, and several limb types meeting the conditions are obtained.