Partnerships for Energy Security and Innovation Act of 2021 This bill requires the Department of Energy (DOE) to establish the Foundation for Energy Security and Innovation. It also authorizes the establishment of the National Energy Technology Laboratory. The Foundation for Energy Security and Innovation, a nonprofit corporation, must support DOE's mission and increase private and philanthropic sector investments that support efforts concerning the development, deployment, and commercialization of innovative technologies that address crosscutting national energy challenges. The foundation may award fellowships and grants for activities relating to research, development, demonstration, maturation, or commercialization of energy and other technologies supported by DOE. In addition, the National Energy Technology Laboratory may establish a nongovernmental Federal Laboratory-Associated Foundation to advance U.S. energy security as specified by the bill.