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Network Function Virtualization (NFV) carries the potential for on-demand deployment of network algorithms in virtual machines (VMs). In large clouds, however, VM resource allocation incurs delays that hinder the dynamic scaling of such NFV deployment. Parallel resource management is a promising direction for boosting performance, but it may significantly increase the communication overhead and the decline ratio of deployment attempts. Our work analyzes the performance of various placement algorithms and provides empirical evidence that state-of-the-art parallel resource management dramatically increases the decline ratio of deterministic algorithms but hardly affects randomized algorithms. We, therefore, introduce APSR -- an efficient parallel random resource management algorithm that requires information only from a small number of hosts and dynamically adjusts the degree of parallelism to provide provable decline ratio guarantees. We formally analyze APSR, evaluate it on real workloads, and integrate it into the popular OpenStack cloud management platform. Our evaluation shows that APSR matches the throughput provided by other parallel schedulers, while achieving up to 13x lower decline ratio and a reduction of over 85% in communication overheads.