The purpose of this work is to study a parallel two-grid algorithm with residual splitting for the solution of partial differential equations. In standard Multigrid (MG) schemes, if the correction on the two levels is computed in parallel, the low frequency components of the error are corrected twice, which may generate inconsistency and thus loss in efficiency. In the parallel MG method proposed by Chan-Tuminaro \citeCT,TU, a residual splitting is then used to avoid this problem. This is the initial point of our study. In this report, we present an abstract analysis of a parallel MG solver based on residual splitting. In particular, we give a convergence proof based on a smoothing property and an approximation property like assumptions. In this approach, a filtering of the correction is done on the fine level. We also show examples of application of such an algorithm. It turns out that a method without correction filtering seems to be more efficient.