We report the experimental demonstration of a continuous-wave all-fiber optical parametric amplifier in the 1 µm band with a record bandwidth of more than 20 THz and a peak gain of almost 40 dB. This is achieved by using a photonic crystal fiber with a high figure of merit and strongly reduced longitudinal dispersion fluctuations. Due to their unique bandwidth and gain characteristics, fiber parametric amplifiers at 1 µm provide an interesting alternative to solid-state or ytterbium-doped fiber amplifiers for ultrafast optical pulse and signal processing.