According to the new condition of the pump peak power, we report a developed approach to achieve time-frequency mapping (TFM) in the spectral Fraunhofer regime by a single all-optical time lens using parabolic cross-phase modulation (XPM). The condition is obtained by combining the far-field spectral Fraunhofer regime with the ability of parabolic pump pulses to induce a pure quadratic phase modulation. The derived condition allows the tunability of the mapping by different strategies when probe pulses with different durations are targeted. For design requirements, we show the complete family of bright and dark parabolic pump pulses, which can be implemented depending on the shape of the probe. In order to overcome the limits of the far-field TFM, we demonstrate as a solution the so-called near-field TFM for narrowband mapping. We also discuss potential applications in a variety of fields to provide more insight into the developed approach