In this paper we develop simple and fast multicore parallel algorithms for counting the number of k-cliques in large undirected graphs, for any small constant k ≥ 4. Clique counting is an important problem in a variety of network analytics applications. Differently from existing solutions, which mainly target distributed memory settings (e.g., MapReduce), our algorithms work on off-the-shelf shared-memory multicore platforms. We assess the effectiveness of our approach through an extensive experimental analysis on a variety of real-world graphs, considering different clique sizes and scalability on different numbers of cores. The experimental results show that our parallel algorithms largely outperform the running times of highly optimized sequential solutions and gracefully scale to non-trivial values of k even on medium/large graphs. For instance, computing hundreds of billions of cliques for rather demanding Web graphs and social networks requires about 15 min on a 32-core machine. As a by-product of our experimental analysis, we also compute the exact number of k-cliques with at most 20 nodes in many real-world networks from the SNAP repository.