This paper describes a novel sensorless force estimation algorithm for the rigid link parallel wrist mechanism of a robotic surgical instrument. The method utilizes novel reaction force observers (RFOB) in joint space, which are modified disturbance observers (DOB) combined with Neural Networks (NN) for inverse dynamics calculations, to estimate external forces acting on the motors. External force/torque estimation in Cartesian space is achieved by the use of the robot Jacobian. The proposed algorithm is applicable to any back-drivable rigid-link wrist mechanism without the need for force sensors. In this paper, the method is implemented on a novel 3 degree-of-freedom (DOF) parallel robotic surgical wrist mechanism that is designed for high dexterity (+/- 90 degrees pitch-yaw rotations, thrust motion) and force/torque estimation. The wrist is actuated extracorporally with 3 rigid push-pull rods and 3 linear motors. With a rigid transmission and high back-drivability, external force/torque estimation can be achieved from the motor position readings utilizing the proposed method. Several experiments were performed on the manufactured prototype of the instrument and results validate the efficacy of the wrist and estimation method with RMS force/torque estimation error values of 0.0024 Nm in pitch axis, 0.0043 Nm in yaw axis and 0.1866 N in thrust axis.