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Multipacting discharge conditions in a parallel geometry are first reconsidered according to the so‐called constant‐k theory of Gill and von Engel [Proc. R. Soc. London A 192, 446 (1948)]. Four dimensionless variables are introduced: two for representing in a normalized form the peak value of the applied voltage and its frequency, two for characterizing the emission of secondary electrons from the electrodes. Numerical solutions of the equations developed by the author in a previous article [J. Appl. Phys. 71, 4629 (1992)] show the existence, when the constant‐k postulate is valid, of a larger than previously known set of possible multipacting resonance modes. The effects on them of the electron phase‐focusing requirements for a stable discharge are outlined. A procedure for determining breakdown voltages of the multipacting discharge through Monte Carlo simulations is then described. A set of simulation results is presented and compared with the corresponding data obtained from the constant‐k theory: Except for the cutoff frequency values, the agreement is very good for the lowest‐mode breakdown. In the presence of higher modes, breakdown voltages obtained from simulations are higher than the corresponding minimum breakdown values from the constant‐k theory, but the transitions between successive modes, when the frequency is increased, can generally be recognized. These results confirm also the main features which are found in the plots of the experimental data reported by various authors, and particularly, at the highest frequencies, the linear dependence of the breakdown voltage on the product of frequency times electrode separation. In addition to breakdown voltages, Monte Carlo simulations provide the corresponding distributions of electron crossing times. These distributions prove the existence, under well‐simulated physical conditions, of the breakdown mode first reported in the author’s - previous article. © 1995 American Institute of Physics.