Incorporating time-varying elements into electromagnetic systems has been shown to be a powerful approach to challenge well-established performance limits, for example, bounds on absorption and impedance matching. So far, most of these studies have concentrated on time-switched systems, where the material undergoes instantaneous modulation in time while the input field is entirely contained within it. This approach, however, necessitates accurate timing of the switching event and limits how thin the system can ultimately be because of the spatial width of the impinging pulse. To address these challenges, here we investigate the periodic temporal modulation of highly lossy materials, focusing on their relatively unexplored parametric absorption aspects. Our results reveal that, by appropriate selection of the modulation parameters, the absorption performance of a periodically modulated absorber can be greatly improved compared with that of its time-invariant counterpart, and can even exceed the theoretical bound for conventional electromagnetic absorbers, namely, the ``Rozanov bound.'' Our findings thus demonstrate the potential of periodic temporal modulations to enable significant improvements in absorber performance while circumventing the limitations imposed by precise timing and material thickness in time-switched schemes, opening up new opportunities for the design and optimization of advanced electromagnetic absorber systems for various applications.