The task of parsing subcutaneous vessels in clinical images is often hindered by the high cost and limited availability of ground truth data, as well as the challenge of low contrast and noisy vessel appearances across different patients and imaging modalities. In this work, we propose a novel weakly supervised training framework specifically designed for subcutaneous vessel segmentation. This method utilizes low-cost, sparse annotations such as centerline traces, dot markers, or short scribbles to guide the learning process. These sparse annotations are expanded into dense probabilistic supervision through a differentiable random walk label propagation model, which integrates vesselness cues and tubular continuity priors driven by image data. The label propagation process results in per-pixel hitting probabilities and uncertainty estimates, which are incorporated into an uncertainty-weighted loss function to prevent overfitting in ambiguous areas. Notably, the label propagation model is trained jointly with a CNN-based segmentation network, allowing the system to learn vessel boundaries and continuity constraints without the need for explicit edge supervision. Additionally, we introduce a topology-aware regularizer that encourages centerline connectivity and penalizes irrelevant branches, further enhancing clinical applicability. Our experiments on clinical subcutaneous imaging datasets demonstrate that our approach consistently outperforms both naive sparse-label training and traditional dense pseudo-labeling methods, yielding more accurate vascular maps and better-calibrated uncertainty, which is crucial for clinical decision-making. This method significantly reduces the annotation workload while maintaining clinically relevant vessel topology.