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Semi-competing risks refers to the survival analysis setting where the occurrence of a non-terminal event is subject to whether a terminal event has occurred, but not vice versa. Semi-competing risks arise in a broad range of clinical contexts, with a novel example being the pregnancy condition preeclampsia, which can only occur before the `terminal' event of giving birth. Models that acknowledge semi-competing risks enable investigation of relationships between covariates and the joint timing of the outcomes, but methods for model selection and prediction of semi-competing risks in high dimensions are lacking. Instead, researchers commonly analyze only a single or composite outcome, losing valuable information and limiting clinical utility -- in the obstetric setting, this means ignoring valuable insight into timing of delivery after preeclampsia has onset. To address this gap we propose a novel penalized estimation framework for frailty-based illness-death multi-state modeling of semi-competing risks. Our approach combines non-convex and structured fusion penalization, inducing global sparsity as well as parsimony across submodels. We perform estimation and model selection via a pathwise routine for non-convex optimization, and prove the first statistical error bound results in this setting. We present a simulation study investigating estimation error and model selection performance, and a comprehensive application of the method to joint risk modeling of preeclampsia and timing of delivery using pregnancy data from an electronic health record.