When solving Hamiltonian systems numerically, it is essential to preserve the symplectic structure of the flow map. Symplectic numerical integrators help with long-term energy conservation and stability. Different approaches to symplectic structure-preserving neural networks have been developed. However, no symplectic neural networks have been developed for parameter-dependent Hamiltonian systems, so far. We propose a neural network architecture based on the generalized Hamiltonian neural networks that can learn a parameter-dependent flow map, while preserving the symplectic structure. We compare these neural networks against physics-unaware multilayer perceptrons, for which integrating a parameter dependency can be easily achieved by adding an additional input neuron. The parameterized generalized Hamiltonian neural networks are able to extrapolate to areas of the phase space without training data, while the physics-unaware multilayer perceptrons are not. Furthermore, they can learn the underlying dynamics with very sparse data compared to the multilayer perceptrons. The parameterized generalized Hamiltonian neural networks achieve all of this with the same prediction speed as multilayer perceptrons.