Engineering plastics have specific properties in the strength, hardness, impact resistance, and aging persistence, often used for structural plates and electronic components. However, the holes made by the drilling process always shrink after the cutting heat dispersion due to their high thermal expansion coefficient. Especially for small-hole fabrication, drilling parameters must be discussed thoughtfully to acquire a stable hole quality. This study developed parameter models by the Taguchi-based neural network method to save the experimental resources on drilling of engineering plastic, polyetheretherketone (PEEK). A three-level full-factorial orthogonal array experiment, L27, was first conducted for minimizing the thrust force, hole shrinkage in diameter, and roundness error. The experiments were operated by a peck-drilling process with cyclic lubricant, and the diameter was 1 mm. In terms of the network modeling, four variables were designated to the input layer neurons including the spindle speed, depth of peck-drilling, feed rate, and thrust force detected; and that of the output layer were the diameter shrinkage and roundness of the hole drilled. The models were trained by a stepped-learning procedure to expand the network’s field information stage by stage. After three stages of training, the models developed can provide precise simulations for the network’s training sets and accurately predict the hole’s characteristics for the non-trained cases.