We consider a continuum model for collective cell movement. Each cell is modelled by a phase field active polar gel model and the cells interact via steric interactions. We provide a finite element implementation with a parallel efficiency of at least 0.5 in the number of cells. This is achieved by considering each cell on a different processor and various improvements to reduce the communication overhead to deal with the cell-cell interactions. We describe implementation details and demonstrate results for up to 768 cells.