Promoting Auto Recalls Toward Safety Act of 2020 or the PARTS Act of 2020 This bill directs the Department of Transportation to establish a program to provide grants and other assistance, including technical assistance, to states for use in notifying motor vehicle owners and lessees about open recalls on their motor vehicles. Additionally, the bill (1) sets forth additional notice requirements for manufacturers regarding motor vehicle or equipment defects and noncompliance with safety standards, and (2) requires the National Highway Traffic Safety Administration to publish an annual scorecard on how effectively manufacturers are completing recalls.