Mobile Edge Computing (MEC) has emerged as a promising paradigm catering to overwhelming explosions of mobile applications, by offloading the compute-intensive tasks to an MEC network for processing. The surging of deep learning brings new vigor and vitality to shape the prospect of intelligent Internet of Things (IoT), and edge intelligence arises to provision real-time deep neural network (DNN) inference services for users. In this paper, we study a novel delay-aware DNN inference throughput maximization problem by accelerating each DNN inference through jointly exploring DNN partitioning and multi-thread parallelism. Specifically, we consider the problem under both offline and online request arrival settings: a set of DNN inference requests is given in advance, and a sequence of DNN inference requests arrives one by one without the knowledge of future arrivals, respectively. We first show that the defined problems are NP-hard. We then devise a novel constant approximation algorithm for the problem under the offline setting. We also propose an online algorithm with a provable competitive ratio for the problem under the online setting. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising.