An eyegaze interface is one of the key technologies that serves as an input device in a ubiquitous-computing society. Recently, video-based techniques that do not require specific instruments have been studied. With these approaches, development of an accurate iris-extraction algorithm is very important to realize practical eyegaze tracking. For accurate iris extraction, it is necessary to achieve robustness, high speed, and high accuracy. Conventional iris-extraction algorithms experience difficulties in meeting all these requirements simultaneously. This study proposes an iris-extraction algorithm based on the parametric template matching method to satisfy all these requirements at the same time. The parametric template matching method achieves robustness by interpolating among some templates, and the method attains high accuracy by a subpixel matching technique. High-speed matching can be realized by coarse-to-fine matching. To verify the effectiveness of the proposed algorithm, we performed a basic experiment for eyegaze tracking. We show in this experiment that the processing time is approximately 1/900 of that of our previous method and that accuracy is acceptable with the new method. Then, we apply the proposed algorithm to an eyegaze keyboard, along with an imaging system for improving image quality, and we verify the effectiveness of this approach.