When multiple LLM coding agents share a rate-limited API endpoint, they exhibit resource contention patterns analogous to unscheduled OS processes competing for CPU, memory, and I/O. In a motivating incident, 3 of 11 parallel agents died from connection resets and HTTP 502 errors - a 27% failure rate - despite the API having sufficient aggregate capacity to serve all 11 sequentially. We present HIVEMIND, a transparent HTTP proxy that applies five OS-inspired scheduling primitives - admission control, rate-limit tracking, AIMD backpressure with circuit breaking, token budget management, and priority queuing - to eliminate the failure modes caused by uncoordinated parallel execution. The proxy requires zero modifications to existing agent code and supports Anthropic, OpenAI, and local model APIs via auto-detected provider profiles. Our evaluation across seven scenarios (5-50 concurrent agents) shows that uncoordinated agents fail at 72-100% rates under contention, while HIVEMIND reduces failures to 0-18% and eliminates 48-100% of wasted compute. An ablation study reveals that transparent retry - not admission control - is the single most critical primitive, but the primitives are most effective in combination. Real-world validation against Ollama confirms that HIVEMIND adds under 3ms of proxy overhead per request. The system is open-source under the MIT license.