Photon pairs produced by the pulse-pumped nonlinear parametric processes have been a workhorse of quantum information science. Engineering the spectral property of the photon pairs is crucial in practical applications. In this article, we demonstrate a programmable photon pair source by exploiting a two-stage nonlinear interferometer with a phase-control device. The phase-control device introduces phase shifts by a programmable phase function that can be arbitrarily defined. With a properly designed phase function, the output spectrum of the source can be freely customized and changed without replacing any hardware component in the system. In addition to demonstrating the generation of photon pairs with factorable, positively-correlated, and negatively-correlated spectra, respectively, we show that the output of the source can be tailored into multi-channel spectrally factorable photon pairs without sacrificing efficiency. Such a source, having the ability to modify the spectrum of the photon pairs at will according to the chosen application, is a powerful tool for quantum information science.