Dynamical behavior of the Duffing’s system with a slowly varying parametric excitation is investigated in this paper. Periodic delayed pitchfork bifurcation behaviors may take place, since the slow parametric excitation can periodically pass through the pitchfork bifurcation point. The dynamical characteristics of bifurcation delay behaviors, especially the resultant bursting oscillations, are discussed. Bifurcation delay behaviors result in hysteresis loops between the stable slow submanifolds, which are responsible for bursting oscillations observed in the parametrically excited Duffing’s system. Furthermore, the effect of initial time on the delay time of each delay behavior is analyzed. The result shows that, given enough time, initial time has no influence on the delay time of the delay behavior, although the delay time of the first delay behavior is decided by initial time.