Stereo vision systems have explored optical coding to enhance depth perception by simultaneously promoting disparity and defocus cues. For instance, to extend the depth of field of stereo cameras, one strategy involves learning phase masks to balance exposure time and light efficiency. However, phase masks are expensive and require high manufacturing precision to avoid unwrapping problems. Inspired by the advantages of coded apertures (CAs) in monocular systems, CAs have also been integrated into the stereo vision camera as a low-cost optical coding solution to improve depth estimation at the expense of light efficiency. In contrast with current CA-based stereo systems that employ CAs originally tailored to monocular systems, we propose jointly learning the CAs and the algorithm for depth estimation in this work. We demonstrate in simulation results that learning a binary or color-coded aperture achieves similar or superior performance to state-of-the-art approaches. Additionally, we validate our findings by building a low-cost prototype using binary CAs made from transparent films, demonstrating the effectiveness of our design in real-world scenarios. Our research contributes to advancing stereo vision technology and highlights the potential of learning-based approaches in optimizing optical coding solutions for depth estimation.