Machine translation (MT) systems universally degrade when faced with code-mixed text. This problem is more acute for low-resource languages that lack dedicated parallel corpora. This work directly addresses this gap for Vietnamese-English, a language context characterized by challenges including orthographic ambiguity and the frequent omission of diacritics in informal text. We introduce VietMix, the first expert-translated, naturally occurring parallel corpus of Vietnamese-English code-mixed text. We establish VietMix's utility by developing a data augmentation pipeline that leverages iterative fine-tuning and targeted filtering. Experiments show that models augmented with our data outperform strong back-translation baselines by up to +3.5 xCOMET points and improve zero-shot models by up to +11.9 points. Our work delivers a foundational resource for a challenging language pair and provides a validated, transferable framework for building and augmenting corpora in other low-resource settings.