Upholding the Law at Our Border Act This bill requires the Office of Inspector General of the Department of Homeland Security (DHS) to periodically investigate and report on certain issues until there have been fewer than 35,000 apprehensions at the southwest border for three consecutive months. When required, the reports must address certain issues, including (1) the vetting procedures applicable to aliens seeking entry into the United States who were apprehended along the southwest border, (2) the total number of individuals not lawfully present who have been processed and released into the United States, (3) the number of such individuals who have received parole, and (4) an audit of the parole applications. The office must submit such reports (and provide briefings on the reports) to the President, Congress, DHS, and the Department of Justice.