We present the foundations for a new dynamic information flow control (IFC) parallel runtime system, LIOPAR. To our knowledge, LIOPAR is the first dynamic language-level IFC system to (1) support parallel thread execution and (2) eliminate both internal- and external-timing covert channels that exploit the runtime system. Most existing IFC systems are vulnerable to external timing attacks because they are built atop vanilla runtime systems that do not account for security—these runtime systems allocate and reclaim shared resources, e.g., CPU-time and memory, fairly between threads at different security levels. While such attacks have largely been ignored—or, at best, mitigated—we demonstrate that extending IFC systems with parallelism leads to the internalization of these attacks. Our IFC runtime system design addresses these concerns by hiearchically managing resources— both CPU-time and memory—and making resource allocation and reclamation explicit at the language-level. We prove that LIOPAR is secure, i.e., it satisfies progress-and-time sensitive non-interference, even when exposing clock and heap-statistics APIs.