As large language models power autonomous agents, understanding how they collaborate and where they fail becomes critical. We introduce PARLEY, an extensible platform for studying LLM agent collaboration through four asymmetric puzzle tasks grounded in cognitive psychology (referential communication, joint coordination, perspective-taking, and spatial reasoning). GPT-4o agent pairs succeeded in 60.5% of conversations (N=400, 100 per task), with task-level variation (34-80%) reflecting distinct collaborative demands. From these conversations, we derived the data-driven PAC taxonomy: six failure types organized into Perception (visual interpretation and spatial reasoning), Adaptation (progress tracking and feedback integration), and Coordination (verification and grounding). Failure to incorporate external feedback was a critical indicator of unsuccessful collaboration, with agents repeatedly applying unsuccessful strategies. We also show that a taxonomy-guided intervention (N=200) improved average task success to 70.5%. These results demonstrate how PARLEY can be used to derive meaningful, data-driven insights that can in turn lead to improved agentic collaboration.

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