Natural Gas Parity Act This bill modifies several tax provisions and federal programs related to the sale, purchase, and use of natural gas. The bill modifies tax provisions related to natural gas to extend the alternative fuels tax credit, modify the excise tax on heavy trucks and trailers sold at retail to add a partial exclusion for alternative fuel trucks, add a new tax credit for new natural gas motor vehicles, extend and increase the alternative fuel vehicle refueling property tax credit, and establish a liquefied natural gas equivalent for the purposes of the Inland Waterways Trust Fund financing rate. The bill modifies several federal programs and requirements to specify minimum federal acquisition requirements for natural gas and alternative fuel vehicles; authorize the Department of Energy (DOE) to conduct specified research, development, and demonstration activities regarding natural gas vehicles; authorize DOE to award natural gas vehicle deployment grants under the Clean Cities program; and require specified funds provided for under certain settlement agreements for alleged violations of environmental laws to be used for alternative fuel vehicle deployment grants.