We present PULP-NN, a multicore computing library for a parallel ultra-low-power cluster of RISC-V based processors. The library consists of a set of kernels for Quantized Neural Network (QNN) inference on edge devices, targeting byte and sub-byte data types, down to INT-1. Our software solution exploits the digital signal processing (DSP) extensions available in the PULP RISC-V processors and the cluster's parallelism, improving performance by up to 63x with respect to a baseline implementation on a single RISC-V core implementing the RV32IMC ISA. Using the PULP-NN routines, the inference of a CIFAR-10 QNN model runs in 30x and 19.6x less clock cycles than the current state-of-the-art ARM CMSIS-NN library, running on an STM32L4 and an STM32H7 MCUs, respectively. By running the library kernels on the GAP-8 processor at the maximum efficiency operating point, the energy efficiency on GAP-8 is 14.1x higher than STM32L4 and 39.5x than STM32H7.