We propose a neural network architecture and a training procedure to estimate blurring operators and deblur images from a single degraded image. Our key assumption is that the forward operators can be parameterized by a low-dimensional vector. The models we consider include a description of the point spread function with Zernike polynomials in the pupil plane or product-convolution expansions, which incorporate space-varying operators. Numerical experiments show that the proposed method can accurately and robustly recover the blur parameters even for large noise levels. For a convolution model, the average signal-to-noise ratio of the recovered point spread function ranges from 13 dB in the noiseless regime to 8 dB in the high-noise regime. In comparison, the tested alternatives yield negative values. This operator estimate can then be used as an input for an unrolled neural network to deblur the image. Quantitative experiments on synthetic data demonstrate that this method outperforms other commonly used methods both perceptually and in terms of SSIM. The algorithm can process a 512 $ \times $ 512 image under a second on a consumer graphics card and does not require any human interaction once the operator parameterization has been set up.1