We build wavelet-based adaptive numerical methods for the simulation of advection-dominated flows that develop multiple spatial scales, with an emphasis on fluid mechanics problems. Wavelet-based adaptivity is inherently sequential and in this work we demonstrate that these numerical methods can be implemented in software that is capable of harnessing the capabilities of multi-core architectures while maintaining their computational efficiency. Recent designs in frameworks for multi-core software development allow us to rethink parallelism as task-based, where parallel tasks are specified and automatically mapped onto physical threads. This way of exposing parallelism enables the parallelization of algorithms that were considered inherently sequential, such as wavelet-based adaptive simulations. In this paper we present a framework that combines wavelet-based adaptivity with the task-based parallelism. We demonstrate the promising performance obtained by simulating various physical systems on different multi-core architectures using up to 16 cores.