The parareal Schwarz waveform relaxation algorithm is a new space-time parallel algorithm for the solution of evolution partial differential equations. It is based on a decomposition of the entire domain both in space and in time into smaller space-time subdomains, and then computes by an iteration in parallel on all these small subdomains a better and better approximation of the overall solution. The initial conditions in the subdomains are updated using a parareal mechanism, while the boundary conditions are updated using Schwarz waveform relaxation techniques. In this talk, an accurate convergence estimate is presented when the algorithm is applied to the heat equation  with the Dirichlet transmission conditions and optimized transmission conditions. We illustrate our analysis with numerical experiments. In addition, a new two level space-time Schwarz waveform relaxation method for time-dependent problems will also be introduced.