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We demonstrate a 40-GHz all-optical time-domain Fourier transformation, in which an ideal parabolic phase modulation is applied to a signal by cross-phase modulation (XPM) with a dark parabolic pulse. The dark optical parabolic pulse is generated by the line-by-line spectral manipulation of a 40-GHz picosecond optical pulse source using an arrayed waveguide grating pulse shaper. By applying the parabolic XPM, the time interval for eliminating linear waveform distortions is expanded to a full bit slot. Dispersion slope-induced distortion is successfully eliminated with this scheme.