In this study, we primarily investigate the asymptotic behavior of solutions associated with a nonclassical diffusion process by memory effects and a perturbed parameter that varies over time. A significant innovation is the consideration of a delay term governed by a function with minimal assumptions: merely measurability and a phase-space that is a time-dependent space of continuously-time-varying functions. By employing a novel analytical approach, we demonstrate the existence and regularity of time-varying pullback $\mathcal{D}$-attractors. Notably, the nonlinearity $f$ is unrestricted by any upper limit on its growth rate.