Nearest Neighbor Search (NNS) has recently drawn a rapid increase of interest due to its core role in managing high-dimensional vector data in data science and AI applications. The interest is fueled by the success of neural embedding, where deep learning models transform unstructured data into semantically correlated feature vectors for data analysis, e.g., recommend popular items. Among several categories of methods for fast NNS, similarity graph is one of the most successful algorithmic trends. Several of the most popular and top-performing similarity graphs, such as NSG and HNSW, at their core employ best-first traversal along the underlying graph indices to search near neighbors. Maximizing the performance of the search is essential for many tasks, especially at the large-scale and high-recall regime. In this work, we provide an in-depth examination of the challenges of the state-of-the-art similarity search algorithms, revealing its challenges in leveraging multi-core processors to speed up the search efficiency. We also exploit whether similarity graph search is robust to deviation from maintaining strict order by allowing multiple walkers to simultaneously advance the search frontier. Based on our insights, we propose Speed-ANN, a parallel similarity search algorithm that exploits hidden intra-query parallelism and memory hierarchy that allows similarity search to take advantage of multiple CPU cores to significantly accelerate search speed while achieving high accuracy. We evaluate Speed-ANN on a wide range of datasets, ranging from million to billion data points, and show its shorter query latency than NSG and HNSW, respectively. Besides, with multicore support, we show that our approach offers faster search latency than highly-optimized GPU implementation and provides good scalability as the increase of the number of hardware resources (e.g., CPU cores) and graph sizes.