Using eye tracking for the investigation of visual attention has become increasingly popular during the last few decades. Nevertheless, only a small number of eye tracking studies have employed 3D displays, although such displays would closely resemble our natural visual environment. Besides higher cost and effort for the experimental setup, the main reason for the avoidance of 3D displays is the problem of computing a subject's current 3D gaze position based on the measured binocular gaze angles. The geometrical approaches to this problem that have been studied so far involved substantial error in the measurement of 3D gaze trajectories. In order to tackle this problem, we developed an anaglyph-based 3D calibration procedure and used a well-suited type of artificial neural network—a parametrized self-organizing map (PSOM)—to estimate the 3D gaze point from a subject's binocular eye-position data. We report an experiment in which the accuracy of the PSOM gaze-point estimation is compared to a geometrical solution. The results show that the neural network approach produces more accurate results than the geometrical method, especially for the depth axis and for distant stimuli.