The parareal in time algorithm allows for efficient parallel numerical simulations oftime-dependent problems. It is based on a decomposition of the time interval intosubintervals, and on a predictor-corrector strategy, where the propagations over eachsubinterval for the corrector stage are concurrently performed on the different processorsthat are available. In this article, we are concerned with the long time integration ofHamiltonian systems. Geometric, structure-preserving integrators are preferably employedfor such systems because they show interesting numerical properties, in particularexcellent preservation of the total energy of the system. Using a symmetrization procedureand/or a (possibly also symmetric) projection step, we introduce here several variants ofthe original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E66 (2002) 057701; G. Bal and Y. Maday, A parareal timediscretization for nonlinear PDE’s with application to the pricing of an American put, inRecent developments in domain decomposition methods, Lect.Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Madayand G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)661–668.] that are better adapted to the Hamiltonian context. These variants arecompatible with the geometric structure of the exact dynamics, and are easy to implement.Numerical tests on several model systems illustrate the remarkable properties of theproposed parareal integrators over long integration times. Some formal elements ofunderstanding are also provided.