  New families of time-dependent potentials related to the parametric
oscillator are introduced. This is achieved by introducing some general
time-dependent operators that factorize the appropriate constant of motion
(quantum invariant) of the parametric oscillator, leading to new families of
quantum invariants that are almost-isospectral to the initial one. Then, the
respective time-dependent Hamiltonians are constructed, and the solutions of
the Schr\"odinger equation are determined from the intertwining relationships
and by finding the appropriate time-dependent complex-phases of the
Lewis-Riesenfeld approach. To illustrate the results, the set of parameters of
the new potentials are fixed such that a family of time-dependent rational
extensions of the parametric oscillator is obtained. Moreover, the rational
extensions of the harmonic oscillator are recovered in the appropriate limit.
