During amplification in a noncollinear optical parametric amplifier the spatial and temporal coordinates of the amplified field are inherently coupled. These couplings or distortions can limit the peak intensity, among other things. In this work, a numerical study of the spatiotemporal distortions in BBO-based noncollinear optical parametric chirped-pulse amplifiers (NOPCPAs) is presented for a wide range of parameters and for different amplification conditions. It is shown that for Gaussian pump beams, gain saturation introduces strong distortions and high conversion efficiency always comes at the price of strong spatiotemporal couplings which drastically reduce the peak intensity even when pulse fronts of the pump and the signal are matched. However, high conversion efficiencies with minimum spatiotemporal distortions can still be achieved with flat-top pump beam profiles.