A 2-DOF (degrees-of-freedom) 3-4R parallel manipulator (PM) with orthogonal base and platform is a novel PM with multiple operation modes that uses the minimum number of actuated joints. A set of constraint equations of the 3-4R PM is derived with the orientation of the moving platform represented using a Euler parameter (also Euler–Rodrigues parameter) quaternion quaternion and then solved using the algebraic geometry method. The motion characteristics of the moving platform are obtained using the kinematic interpretation of Euler parameter quaternions with certain number of constant zero components. It is found that this 3-4R PM has 14 2-DOF operation modes, including four spherical translation modes, six planar motion modes, and four sphere-on-sphere rolling modes. The transition configurations among different operation modes are also presented. The maximum number of operation modes among which the 3-4R PM can switch in the same transition configuration is four. As a by-product, a concise representation of 2-DOF sphere-on-sphere rolling is obtained.