<p>We consider some geometrical aspects associated with the dynamics of two-mode Gaussian states evolving under two different processes: parametric conversion and parametric amplification. The evolved Gaussian states are represented by points of four dimensional manifolds. We employ the Hilbert-Schmidt to measure the distance between two Gaussian states and we determine the Riemannian metric in each case. This study provides a geometrical description of the evolution from separable to entangled two-mode Gaussian states. This allows a geometric description of their evolution under time-dependent parametric interactions. Parametric conversion and amplification are quintessential in the manipulation of quantum states, enabling the transformation and enhancement of their entanglement properties. In particular, we focus on the investigation of the evolution of the entanglement in Gaussian states from purely geometric perspective offering insights how parametric processes impact the manifolds curvature in connection with the entanglement generation.</p>