Scheduling query execution plans is a particularly complex problem in hierarchical parallel systems, where each site consists of a collection of local time-shared (e.g., CPU(s) or disk(s)) and space-shared (e.g., memory) resources and communicates with remote sites by message passing. We develop a general approach to the problem, capturing the full complexity of scheduling distributed multi-dimensional resource units for all kinds of parallelism within and across queries and operators. We present heuristic algorithms for various forms of the problem, some of which are provably near-optimal. Preliminary experimental results confirm the effectiveness of our approach.