The time-optimal control of a classical damped harmonic oscillator takes the form of a periodic step-like modulation of the oscillator stiffness, as established in [1]. Building on this work, we show that its emergent periodic structure admits a direct dynamical interpretation in terms of parametric instability. We further derive new weak-damping asymptotics showing that the optimal frequency associated to a unit amplification approaches twice the natural frequency of the oscillator, revealing a direct and unexpected link between optimality and the condition for parametric resonance. This observation enables a direct mapping between optimal solutions and the stability structure of dynamical systems, as captured by the Pontryagin maximum principle and Floquet theory. Our results reveal a connection between time-optimal control and regimes of strong parametric instability.