Stereo matching in minimally invasive surgery (MIS) is essential for next-generation navigation and augmented reality. Yet, dense disparity supervision is nearly impossible due to anatomical constraints, typically limiting annotations to only a few image-level labels acquired before the endoscope enters deep body cavities. Teacher-Student Learning (TSL) offers a promising solution by leveraging a teacher trained on sparse labels to generate pseudo labels and associated confidence maps from abundant unlabeled surgical videos. However, existing TSL methods are confined to image-level supervision, providing only spatial confidence and lacking temporal consistency estimation. This absence of spatio-temporal reliability results in unstable disparity predictions and severe flickering artifacts across video frames. To overcome these challenges, we propose TiS-TSL, a novel time-switchable teacher-student learning framework for video stereo matching under minimal supervision. At its core is a unified model that operates in three distinct modes: Image-Prediction (IP), Forward Video-Prediction (FVP), and Backward Video-Prediction (BVP), enabling flexible temporal modeling within a single architecture. Enabled by this unified model, TiS-TSL adopts a two-stage learning strategy. The Image-to-Video (I2V) stage transfers sparse image-level knowledge to initialize temporal modeling. The subsequent Video-to-Video (V2V) stage refines temporal disparity predictions by comparing forward and backward predictions to calculate bidirectional spatio-temporal consistency. This consistency identifies unreliable regions across frames, filters noisy video-level pseudo labels, and enforces temporal coherence. Experimental results on two public datasets demonstrate that TiS-TSL exceeds other image-based state-of-the-arts by improving TEPE and EPE by at least 2.11% and 4.54%, respectively.