In this study, we propose a morpheme-based scheme for Korean dependency parsing and adopt the proposed scheme to Universal Dependencies. We present the linguistic rationale that illustrates the motivation and the necessity of adopting the morpheme-based format, and develop scripts that convert between the original format used by Universal Dependencies and the proposed morpheme-based format automatically. The effectiveness of the proposed format for Korean dependency parsing is then testified by both statistical and neural models, including UDPipe and Stanza, with our carefully constructed morpheme-based word embedding for Korean. morphUD outperforms parsing results for all Korean UD treebanks, and we also present detailed error analysis.