We report a high-repetition-rate, compact, mid-infrared picosecond (ps) optical parametric oscillator (OPO) based on periodically poled lithium niobate (PPLN). The OPO is synchronously pumped by an ytterbium-doped-fibre (YDF) master-oscillator-power-amplifier (MOPA) system, seeded by a 1040-nm gain-switched laser diode (GSLD). Under continuous-wave (cw) pulsed operation, at a 1.5-GHz repetition rate and 14-W pump power, an idler average power of 2.4 W (∼30 W peak power) was achieved, with an idler wavelength tunability of 2260-3573 nm. Through the addition of an electro-optic modulator (EOM) to the MOPA system, acting as a time gate to suppress a variable number of pulses per 1 µs, a quasi-cw pumping regime was realized, allowing burst-mode operation of the OPO at a 1-MHz inter-burst repetition rate. By varying the burst window time with the EOM, controllable idler peak powers of up to 1.2 kW were then realized.