Drilling quality is highly related to the fatigue strength of the aircraft components, and high-precision drilling is critical for aircraft. Parallel robots feature superior position and orientation accuracy over serial robots, making them ideal choices for high-precision drilling. To guarantee the drilling flexibly and accuracy, a 5-DoF parallel robot that can achieve flexible translation in 3D space and orientation adjustment is adopted and three laser displacement sensors (LDSs) are installed at the end of the parallel robot to measure the workpiece’s normal. However, the parallel robot’s parasitic motion poses challenges to sensor mounting position calibration. To address this issue, the traditional calibration method with fixed calibration plate and moving robot is modified to fixed robot and moving the calibration plate. Then, the randomized random sample consensus (R-RANSAC) algorithm is employed to derive accurate mounting positions of LDSs. In this way, the light point and the path vector can be derived in the drilling process, and the workpiece’s normal measurement is achieved online. Normal measurement experimental results show that the average measuring error is 0.167° after calibration. Finally, drilling experiments on an aluminum alloy plate are also conducted, and the experimental results show an average hole’s perpendicularity of 0.306° which meet the accuracy requirement of 0.5°. This illustrates the validity of the proposed sensor mounting position calibration and online normal adjustment method.