We report on the first realization of an ultrafast Airy beam optical parametric oscillator (OPO). By introducing intracavity cubic phase modulation to the resonant Gaussian signal in a synchronously-pumped singly-resonant OPO cavity and its subsequent Fourier transformation, we have generated 2-dimensional Airy beam in the output signal across a 250 nm tuning range in the near-infrared. The generated Airy beam can be tuned continuously from 1477 to 1727 nm, providing an average power of as much as 306 mW at 1632 nm in pulses of ~23 ps duration with a spectral bandwidth of 1.7 nm.