In this paper we introduce a response time analysis technique for Synchronous Data Flow programs mapped to multiple parallel dependent tasks running on a compute cluster of the Kalray MPPA-256 many-core processor. The analysis we derive computes a set of response times and release dates that respect the constraints in the task dependency graph. We extend the Multicore Response Time Analysis (MRTA) framework by deriving a mathematical model of the multi-level bus arbitration policy used by the MPPA. Further, we refine the analysis to account for the release dates and response times of co-runners, and the use of memory banks. Further improvements to the precision of the analysis were achieved by splitting each task into two sequential phases, with the majority of the memory accesses in the first phase, and a small number of writes in the second phase. Our experimental evaluation focused on an avionics case study. Using measurements from the Kalray MPPA-256 as a basis, we show that the new analysis leads to response times that are a factor of 4.15 smaller for this application, than the default approach of assuming worst-case interference on each memory access.