We report on Monte Carlo simulations of solar energetic particle (SEP) acceleration at quasi-parallel coronal shocks under the influence of self-generated Alfvén waves. The results indicate that the accelerated particles amplify ambient Alfvén waves efficiently and that the solution close to the shock can be qualitatively described with the results from quasi-steady theories of diffusive shock acceleration, provided that the acceleration and injection parameters do not change rapidly. The escape of the first particles to the interplanetary medium occurs before the waves have grown appreciably to trap the particles in the vicinity of the shock wave. The escape process is well described by the analytical model developed by Vainio, at least for the promptly escaping component. In addition to the compression ratio and speed of the shock wave, the rate of injection of low-energy particles to the acceleration process is a key factor for the acceleration efficiency of shocks that are driven by coronal mass ejection. Quasi-parallel coronal shocks seem to be capable of accelerating suprathermal protons up to 100 MeV and beyond after some number of minutes. Extrapolations of our simulation results indicate, however, that the wave intensities may reach nonlinear values before acceleration to GeV energies occurs in the corona. This may mean that the quasi-linear approach has to be replaced by a more general theory to describe particle acceleration at quasi-parallel coronal shocks in the largest SEP events.