The network switches in the data plane of Software Defined Networking (SDN) are empowered by an elementary process, in which enormous number of packets which resemble big volumes of data are classified into specific flows by matching them against a set of dynamic rules. This basic process accelerates the processing of data, so that instead of processing singular packets repeatedly, corresponding actions are performed on corresponding flows of packets. In this paper, first, we address limitations on a typical packet classification algorithm like Tuple Space Search (TSS). Then, we present a set of different scenarios to parallelize it on different parallel processing platforms, including Graphics Processing Units (GPUs), clusters of Central Processing Units (CPUs), and hybrid clusters. Experimental results show that the hybrid cluster provides the best platform for parallelizing packet classification algorithms, which promises the average throughput rate of 4.2 Million packets per second (Mpps). That is, the hybrid cluster produced by the integration of Compute Unified Device Architecture (CUDA), Message Passing Interface (MPI), and OpenMP programming model could classify 0.24 million packets per second more than the GPU cluster scheme. Such a packet classifier satisfies the required processing speed in the programmable network systems that would be used to communicate big medical data.