In this paper we propose a constant-time parallel algorithm for implementing the message-passing decoder of LDPC codes on a two dimensional RMesh, trying to keep the number of processors small. The R-Mesh provides dynamic reconfiguration, hardware reuse, and flexibility to problem changes. To decode a different code, we may simply set up the required connections between the bit-nodes and check-nodes by modifying the initialization phase of the R-Mesh algorithm. No extra wiring or hardware changes are required, as compared to other existing approaches. Moreover, the same hardware can implement the decoder in both probability and logarithm domains. We illustrate that the R-Mesh is an efficient model for parallel implementation of the decoder in terms of time complexity, flexibility to problem changes and simplicity of routing messages.