In this paper, we address two long-standing challenges in neuro-oncology: (1) how to leverage large amounts of unlabeled magnetic resonance (MR) imaging data for radiogenomic tasks and (2) how to unite glioma MR imaging with genomic data. We examine multi-parametric MR data from 542 patients in the combined training, validation, and testing sets of the 2018 Multimodal Brain Tumor Segmentation Challenge and somatic copy number alteration (SCNA) data from 1090 patients in The Cancer Genome Archive's (TCGA) lower-grade glioma and glioblastoma projects. We propose a novel application of multi-task learning (MTL) that leverages unlabeled MR data by jointly learning tumor segmentation masks with glioma molecular subtype markers and allows for SCNA input when available. There are 235 patients in the intersection of these MR and SCNA datasets, which we divide into an unlabeled training set, a labeled training set, and a validation set. Our MTL model significantly outperforms comparable classification models trained only on labeled MR data for both IDH1/2 mutation and 1p/19q co-deletion glioma subtype marker prediction tasks. We also observe that models trained on genomic and imaging data improve survival prediction results achieved by models trained on either alone. We will release our source code for future research.