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A growing use for serverless computing is large parallel data processing applications that take advantage of its on-demand scalability. Because individual serverless compute nodes, which are called functions, run in isolated containers, a major challenge with this paradigm is transferring temporary computation data between functions. Previous works have performed inter-function communication using object storage, which is slow, or in-memory databases, which are expensive. We evaluate the use of direct network connections between functions to overcome these limitations. Although function containers block incoming connections, we are able to bypass this restriction using standard NAT traversal techniques. By using an external server, we implement TCP hole punching to establish direct TCP connections between functions. In addition, we develop a communications framework to manage NAT traversal and data flow for applications using direct network connections. We evaluate this framework with a reduce-by-key application compared to an equivalent version that uses object storage for communication. For a job with 100+ functions, our TCP implementation runs 4.7 times faster at almost half the cost.