We demonstrate a picosecond fiber-optical parametric oscillator with an ultrawide tuning range pumped by a relatively low-cost intensity-modulated pump. The tuning range of the oscillator is from 1320 to 1520 nm and from 1580 to 1820 nm, which is as wide as 440 nm, with a wavelength span of 500 nm. A 40-m dispersion-shifted fiber is employed inside a cavity as the parametric gain medium. In the wavelength region from 1320 to 1395 nm and from 1720 to 1820 nm, the linewidth of the sideband is less than 1 nm without the use of any wavelength-selective element inside the cavity. Pulses with pulsewidths from 7 to 17 ps are generated.