It happened at Father and Son Moving Co. on Frelinghuysen Avenue.
Parker was employed by the Moving Company, but authorities said he was asked to leave the premises this morning.
Investigators said he left but later returned with a can of gasoline which he used to start a fire in the foyer.
There was minimal damage to the property.
Parker was taken into custody around 1:30 p.m. without incident in the vicinity of Elizabeth Avenue and West Alpine Street by detectives from the Essex County Prosecutor's Office Homicide/Major Crimes Task Force.