Quality Assurance, really Quality Managment, is one aspect of a project. One of PMI's Project Management Discipline areas is quality management, which in terms of execution (not planning) is usually broken down into quality control and quality assurance (QA).
Quality control is concerned with evaluating work products to make sure they conform to specs (eg, testing, in the software world). Quality Assurance is about making sure that the business processes that are understood to produce a goodproduct are followed -- it's focused on processes, rather than product. ISO 9000 recognizes this concept, also. If you know, for example, that peer code review improves the quality of your software products, then Quality Assurance is concerned with making sure that everyone knows how to follow the company/project standards for peer review and that they are in fact followed. Similarly, QA would be responsible for making sure the processes remain effective and recommending changes to keep them that way.
Generally, QA tasks are undertaken by line managers, team leads and, to some extent, a project manager. QA is usually overseen and directed by a dedicated QA person, usually a senior manager who reports directly to the person most accountable for the quality of the product in question (in your example, the Project Manager).
Note that the term QA is often misused. Most correctly, QA is involved in processes, and the NOT testing of products. Often, testing is incorrectly referred to as QA. However, the testing function is often simlilarly arrange, with a test manager (or QC manager) reporting to the Project Manager. The Project Manager should be concerned both with QA and QC.
PMI's Project Management Body of Knowledge offers an approachable discussion of the topic, and there are numerous books available, as well.