  We present a Lie algebraic approach to a Hamiltonian class covering driven,
parametric quantum harmonic oscillators where the parameter set -- mass,
frequency, driving strength, and parametric pumping -- is time-dependent. Our
unitary-transformation-based approach provides a solution to our general
quadratic time-dependent quantum harmonic model. As an example, we show an
analytic solution to the periodically driven quantum harmonic oscillator
without the rotating wave approximation; it works for any given detuning and
coupling strength regime. For the sake of validation, we provide an analytic
solution to the historical Caldirola--Kanai quantum harmonic oscillator and
show that there exists a unitary transformation within our framework that takes
a generalized version of it onto the Paul trap Hamiltonian. In addition, we
show how our approach provides the dynamics of generalized models whose
Schr\"odinger equation becomes numerically unstable in the laboratory frame.
