Block implicit methods (BIM) is a family of time discretization schemes with desirable stability properties, high order of accuracy, and requires a single initial value. Moreover, BIM is also time-parallel since the solutions at several time steps can be computed simultaneously at a time. In this talk, we present a multilevel additive Schwarz preconditioner for solving the large sparse system of algebraic equations arising from the discretization by BIM in time and finite element in space. We prove an optimal convergence theory showing that the convergence rate is bounded independently of the mesh parameters, the number of subdomains and the number of levels. Numerical experiments carried out on a parallel computer with thousands of processors confirm the optimality and scalability of the method.