Code-mixed text from social media poses significant challenges for syntactic analysis due to irregular grammar, non-standard usage, and frequent language switching. For Telugu-English code-mixed text, the absence of large-scale syntactic resources and specialized parsing models limits progress in downstream multilingual NLP applications. In this work, we address this gap by introducing the first substantial manually annotated Telugu-English code-mixed dependency treebank of 4,152 sentences, developed using Universal Dependencies (UD) 2.0 guidelines. We further propose enhancements to a biaffine dependency parser by incorporating a language-aware head-dependent bias and relation-specific structural weights to better capture cross-lingual syntactic patterns. Our approach improves parsing performance, achieving 75.53% UAS and 61.86% LAS, with consistent gains over a strong baseline. In addition, we demonstrate that integrating dependency-derived syntactic features into a BiLSTM-CRF model improves part-of-speech tagging, achieving a macro-F1 score of 83.73%, with statistically validated gains. We also re-annotate an existing Telugu-English dataset using UD 2.0 to ensure compatibility with modern syntactic frameworks. Overall, this work provides new annotated resources and modeling strategies that advance syntactic processing for Telugu-English code-mixed text, with broader implications for developing robust NLP systems in low-resource and multilingual settings.