Decoupled Software Pipelining (DSWP) is a program partitioning method enabling compilers to extract pipeline parallelism from sequential programs. Parallel Stage DSWP (PS-DSWP) is an extension that also exploits the data parallelism within pipeline filters. This paper presents the preliminary design of a new PS-DSWP method capable of handling arbitrary structured control flow, a slightly better algorithmic complexity, the natural exploitation of nested parallelism with communications across arbitrary levels, with a seamless integration with data-flow parallel programming environments. It is inspired by loop-distribution and supports nested/structured partitioning along with the hierarchy of control dependences. The method relies on a data-flow streaming extension of OpenMP. These advances are made possible thanks to progresses in compiler intermediate representation. We describe our usage of the Static Single Assignment (SSA) form, how we extend it to the context of concurrent streaming tasks, and we discuss the benefits and challenges for PS-DSWP.