In this talk we discuss a parallel fully implicit Newton-Krylov-Schwarz algorithm for the numerical solution of the unsteady magnetic reconnection problem described by the system of the reduced magnetohydrodynamics (MHD) equations in two-dimensional space. In the MHD formalism plasma is treated as conducting fluid and behaves according to the fluid dynamics equations, coupled with the system of Maxwell’s equations. One of the intrinsic features of MHD is the formation of singular current density sheets, which is believed to be linked to the reconnecting of magnetic fields. A robust solver is required for handling the high nonlinearity associated with the simulation of magnetic reconnection phenomena. The near singular behavior of the solution of the system often limits the usable time step size required by explicit schemes thus making implicit methods potentially more attractive. We employ a stream function approach to enforce the divergence-free conditions on the magnetic and velocity fields, and solve the resulting fully coupled current-vorticity system of equations with a fully implicit time integration using Newton-Krylov techniques with an one-level additive Schwarz preconditioning. In this work we study the parallel convergence of the implicit algorithm and compare our results with those obtained by an explicit method.