Spoken bilingual conversations pose substantial challenges for syntactic parsing because they often include disfluencies and discourse-driven structures that complicate dependency parsing under standard Universal Dependencies (UD) assumptions and evaluation practices. To systematically study these challenges, in this work, we first introduce a linguistically grounded taxonomy of conversational bilingual phenomena, together with SpokeBench, an expert-annotated English-Spanish benchmark for structurally complex speech. To address the limitations of existing evaluation practices, we propose Flex-UD, an ambiguity-aware evaluation metric that distinguishes catastrophic structural failures from linguistically acceptable variations. Finally, we introduce DECAP, a decoupled agentic parsing framework that separates spoken-phenomena handling from core syntactic analysis, enabling robust and interpretable dependency parsing without retraining. Experiments across both proprietary and open-weight LLMs show that DECAP substantially improves performance on complex conversational phenomena and achieves over 60% improvements in UPOS-F1 Score over baselines, while Flex-UD evaluations reveal gains that otherwise remain partially hidden under standard attachment-based metrics.