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We present an extension of the klystron simulation code TESLA to model multiple-beam klystrons (MBKs) in which interaction parameters may vary significantly from beam-tunnel to beam-tunnel. In earlier work, the single-beam code was applied to model the MBK by assuming that all electron beams and beam-tunnels were identical and all electron beams interacted identically with the fields of the resonant cavities, using averaged values of R/Q to represent interaction with each resonant cavity. To overcome the limitations of this approach and to take into account the effects from nonidentical beams and/or beam-tunnels, we have modified the code to use a parallel algorithm for multiple beams. The implementation of the parallel version of TESLA is based on the latest Fortran-95 version of the serial code and uses the message-passing interface library for communication. For testing and verification purposes, the new version of the code is applied to simulate an experimental four-cavity, eight-beam klystron amplifier, which was designed and successfully tested last year at the Naval Research Laboratory. The results of modeling using the new parallel TESLA and their comparison with experimental data are discussed in detail.