The serverless computing model has been on the rise in recent years due to a lower barrier to entry and elastic scalability. However, our experimental evidence suggests that multiple serverless computing platforms suffer from serious performance inefficiencies when a high number of concurrent function instances are invoked, which is a desirable capability for parallel applications. To mitigate this challenge, this paper introduces ProPack, a novel solution that provides higher performance and yields cost savings for end users running applications with high concurrency. ProPack leverages insights obtained from experimental study to build a simple and effective analytical model that mitigates the scalability bottleneck. Our evaluation on multiple serverless platforms including AWS Lambda and Google confirms that ProPack can improve average performance by 85% and save cost by 66%. ProPack provides significant improvement (over 50%) over the state-of-the-art serverless workload manager such as Pywren, and is also, effective at mitigating the concurrency bottleneck for FuncX, a recent on-premise serverless execution platform for parallel applications.