A conservative approach for MPI-based parallelization of tridiagonal compact schemes is developed in the context of multi-block finite-volume methods. For each block, an enlarged linear system is solved by overlapping a certain number of neighbour cells from adjacent sub-domains. The values at block-to-block boundary faces are evaluated by a high-order centered approximation formula. Unlike previous methods, conservation is retained by properly re-computing the common interface value between two neighbouring blocks. Numerical tests show that parallelization artifacts decrease significantly as the number of overlapping cells is increased, at some expense of parallel efficiency. A reasonable trade-off between accuracy and performances is discussed in the paper with reference to both the spectral properties of the method and the results of fully turbulent numerical simulations.