  The parareal in time algorithm allows to efficiently use parallel computing
for the simulation of time-dependent problems. It is based on a decomposition
of the time interval into subintervals, and on a predictor-corrector strategy,
where the propagations over each subinterval for the corrector stage are
concurrently performed on the processors. In this article, we are concerned
with the long time integration of Hamiltonian systems. Geometric,
structure-preserving integrators are preferably employed for such systems
because they show interesting numerical properties, in particular excellent
preservation of the total energy of the system. Using a symmetrization
procedure and/or a (possibly also symmetric) projection step, we introduce here
several variants of the original plain parareal in time algorithm [Lions, Maday
and Turinici 2001, Baffico, Bernard, Maday, Turinici and Zerah 2002, Bal and
Maday 2002] that are better adapted to the Hamiltonian context. These variants
are compatible with the geometric structure of the exact dynamics, and are easy
to implement. Numerical tests on several model systems illustrate the
remarkable properties of the proposed parareal integrators over long
integration times. Some formal elements of understanding are also provided.
