Using radiological scans to identify liver tumors is crucial for proper patient treatment. This is highly challenging, as top radiologists only achieve F1 scores of roughly 80% (hepatocellular carcinoma (HCC) vs. others) with only moderate inter-rater agreement, even when using multi-phase magnetic resonance (MR) imagery. Thus, there is great impetus for computer-aided diagnosis (CAD) solutions. A critical challenge is to robustly parse a 3D MR volume to localize diagnosable regions of interest (ROI), especially for edge cases. In this paper, we break down this problem using a key-slice parser (KSP), which emulates physician workflows by first identifying key slices and then localizing their corresponding key ROIs. To achieve robustness, the KSP also uses curve-parsing and detection confidence re-weighting. We evaluate our approach on the largest multi-phase MR liver lesion test dataset to date (430 biopsy-confirmed patients). Experiments demonstrate that our KSP can localize diagnosable ROIs with high reliability: 87% patients have an average 3D overlap of >= 40% with the ground truth compared to only 79% using the best tested detector. When coupled with a classifier, we achieve an HCC vs. others F1 score of 0.801, providing a fully-automated CAD performance comparable to top human physicians.