Deep learning has excelled in medical image classification, but its clinical application is limited by poor interpretability. Capsule networks, known for encoding hierarchical relationships and spatial features, show potential in addressing this issue. Nevertheless, traditional capsule networks often underperform due to their shallow structures, and deeper variants lack hierarchical architectures, thereby compromising interpretability. This paper introduces a novel capsule network, ParseCaps, which utilizes the sparse axial attention routing and parse convolutional capsule layer to form a parse-tree-like structure, enhancing both depth and interpretability. Firstly, sparse axial attention routing optimizes connections between child and parent capsules, as well as emphasizes the weight distribution across instantiation parameters of parent capsules. Secondly, the parse convolutional capsule layer generates capsule predictions aligning with the parse tree. Finally, based on the loss design that is effective whether concept ground truth exists or not, ParseCaps advances interpretability by associating each dimension of the global capsule with a comprehensible concept, thereby facilitating clinician trust and understanding of the model's classification results. Experimental results on CE-MRI, PH$^2$, and Derm7pt datasets show that ParseCaps not only outperforms other capsule network variants in classification accuracy, redundancy reduction and robustness, but also provides interpretable explanations, regardless of the availability of concept labels.