Symplecticity-preserving neural networks such as \(\textrm{SympNets}\) have been proposed to learn the flow of symplectic Hamiltonian dynamics and to obtain qualitatively better long-term predictions. Computationally, learning high-dimensional problems still poses a great challenge. Structure-preserving dimensionality reduction methods have been developed to improve computational efficiency, such as proper symplectic decomposition (PSD), to preserve the inherent geometric properties of the system when learning Hamiltonian dynamics. Several near-optimal PSD solutions, such as a cotangent lift solution, have also been constructed. In this work, we propose a symplecticity-preserving unconstrained parametrization of the symplectic lift matrices, such that the dimensionality reduction can be learned simultaneously with learning Hamiltonian dynamics in the dimension-reduced phase space. With this approach, we obtain more accurate numerical results, especially long-term predictions, compared to learning dimension-reduced dynamics with the previously introduced constant PSD cotangent lift solution.