Parametric 3D models have enabled a wide variety of computer vision and graphics tasks, such as modeling human faces, bodies and hands. In 3D face modeling, 3DMM is the most widely used parametric model, but can't generate fine geometric details solely from identity and expression inputs. To tackle this limitation, we propose a neural parametric model named DNPM for the facial geometric details, which utilizes deep neural network to extract latent codes from facial displacement maps encoding details and wrinkles. Built upon DNPM, a novel 3DMM named Detailed3DMM is proposed, which augments traditional 3DMMs by including the synthesis of facial details only from the identity and expression inputs. Moreover, we show that DNPM and Detailed3DMM can facilitate two downstream applications: speech-driven detailed 3D facial animation and 3D face reconstruction from a degraded image. Extensive experiments have shown the usefulness of DNPM and Detailed3DMM, and the progressiveness of two proposed applications.