The United States Justice Department said on Tuesday that it was investigating the death of Victor White III, 23, who died while in the custody of Iberia Parish sheriff’s deputies in March. The local coroner determined that his shooting death in the back of a police cruiser while his hands were cuffed was a suicide. His family has disputed details of the shooting and called for a federal investigation, comparing his death to the fatal police shooting of another young black man, Michael Brown, in Ferguson, Mo. The United States attorney for the Western District of Louisiana, Stephanie A. Finley, said on Tuesday that the F.B.I. had been “working parallel” with the current investigation by the Louisiana State Police, and that the federal investigation would “supplement, rather than supplant,” the state investigation. Officials from the F.B.I., the United States attorney’s office and the Justice Department’s Civil Rights Division will review the results of the investigation and determine whether to investigative further and whether any civil rights violation occurred, Ms. Finley said.