Collision probability evaluation and flux computation are the most time-consuming aspects of applications based on the linearized time-independent transport equation. Parallelization for collision probability calculation and multigroup flux computation are investigated. Particular techniques pertinent to the two-step energy/space iterative process of solving a multigroup transport equation are described. The parallel performance is studied in cases where the cyclic tracking technique is used to integrate collision probability. Parallelization is achieved by distributing either different energy groups or different regions on a set of processors. These algorithms were tested on a four-processor IBM SP2 and an eight-processor SPARC 1000 as well as on networks of workstations using the public domain PVM library. Typical run times are provided for unit cell calculations.