Herein, we present a picosecond-pulsed optical vortex parametric oscillator capable of generating high-power, narrow-linewidth near- and mid-infrared optical vortex outputs. The optical parametric oscillator (OPO), consisting of a KTiOAsO₄ (KTA) crystal and a Z-shaped standing wave cavity formed by five mirrors, transferred the orbital angular momentum (OAM) of the pump field to the signal and idler fields. The transmission mechanism of the OAM within the signal singly resonantsingly-resonant KTA-OPO was investigated, and the OAM was controlled and selectively transferred among the pump, signal, and idler fields by adjusting the focus position of the pump beam on the KTA crystal. With an incident pump power of 17 W, the maximum average output power was 2.14 W at 1535 nm (signal vortex field) and 0.95 W at 3468 nm (idler vortex field), respectively, corresponding to optical conversion efficiencies of 20.8% and 9.2%. The spectral linewidths of the signal and idler vortex fields were 0.502 nm and 1.216 nm, respectively. To the best of our knowledge, this is the first instance of a picosecond-pulsed optical vortex parametric oscillator with a KTA crystal.