The authors have studied numerically the parallel dynamics of nonsymmetric Sherrington-Kirkpatrick spin glasses, varying the degree of symmetry eta :=(Ji, kJk, i)/(Ji, k2) of the coupling coefficients between 0 and 1. For systems of finite size N, in the limit t to infinity and at 'zero temperature', T=0, they find subtle behaviour of the function (C2(t)):=(si(t-1)si (t+1)), which characterizes the appearance of 2-cycles or fixed-point attractors quantitatively. One has to distinguish the two cases eta >0.5, where (C2( infinity )) to 1, i.e. the system is eventually trapped with probability 1 in a fixed-point or a 2-cycle, if after t to infinity the limit N to infinity is taken, and eta <0.5, where, in contrast, (C2( infinity )) is <1, since longer cycles appear. However, the 'trapping' for eta >0.5 happens only for T=0, and at time scales tau N which increase exponentially with N, whereas for T>0, or if for T=0 the limit N to infinity would be taken before t to infinity , the quantity (C2(t to infinity )) would decrease smoothly and monotonically with decreasing eta right from eta =1, in quantitative agreement with the mean-field simulation of Eissfeller and Opper. For T=0, the transient behaviour of (C2(t)) between the mean-field value, which is reached already after typically 100 time steps, and the trapping event, is found to be governed by log-normal statistics with size-depending parameters.