  This paper proposes a parallel in time (called also time parareal) method to
solve Volterra integral equations of the second kind. The parallel in time
approach follows the same spirit as the domain decomposition that consists of
breaking the domain of computation into subdomains and solving iteratively the
sub-problems in a parallel way. To obtain high order of accuracy, a spectral
collocation accuracy enhancement in subdomains will be employed. Our main
contributions in this work are two folds: (i) a time parareal method is
designed for the integral equations, which to our knowledge is the first of its
kind. The new method is an iterative process combining a coarse prediction in
the whole domain with fine corrections in subdomains by using spectral
approximation, leading to an algorithm of very high accuracy; (ii) a rigorous
convergence analysis of the overall method is provided. The numerical
experiment confirms that the overall computational cost is considerably reduced
while the desired spectral rate of convergence can be obtained.
