We consider workflow graphs as a model for the control flow of a business process model and study the problem of workflow graph parsing, i.e., finding the structure of a workflow graph. More precisely, we want to find a decomposition of a workflow graph into a hierarchy of sub-workflows that are subgraphs with a single entry and a single exit of control. Such a decomposition is the crucial step, for example, to translate a process modeled in a graph-based language such as BPMN into a process modeled in a block-based language such as BPEL. For this and other applications, it is desirable that the decomposition be unique, modularand as fine as possible, where modularmeans that a local change of the workflow graph can only cause a local change of the decomposition. In this paper, we provide a decomposition that is unique, modular and finer than in previous work. It is based on and extends similar work for sequential programs by Tarjan and Valdes [11]. We show that our decomposition can be computed in linear time based on an algorithm by Hopcroft and Tarjan [3] that finds the triconnected components of a biconnected graph.