Krylov methods are widely used for solving large sparse linear systems of equations. On distributed architectures, their performance is limited by the communication needed at each iteration of the algorithm. In this paper, we study the use of so-called enlarged Krylov subspaces for reducing the number of iterations, and therefore the overall communication, of Krylov methods. In particular, we consider a reformulation of the conjugate gradient method using these enlarged Krylov subspaces: the enlarged conjugate gradient method. We present the parallel design of two variants of the enlarged conjugate gradient method, as well as their corresponding dynamic versions, where the number of search directions is dynamically reduced during the iterations. For a linear elasticity problem with heterogeneous coefficients, using a block Jacobi preconditioner, we show that this implementation scales up to 16,384 cores and is up to 6.9 times faster than the PETSc implementation of PCG.