Mr. Parker, 59, died this week at his home, shot in the neck and chest and then stabbed. His wife was also shot and stabbed.
Sheriff's officials believe that the double murder in Jonesville, a one-square-mile town of about 900 residents, was not the byproduct of a botched drug deal or a home invasion. Rather, they suspect that Mr. Parker's death was intended as the opening phase of a man's quest to purge sex offenders from Union County.
"He went through our sex offender registry," said Sheriff David H. Taylor, "and individually picked out targets."
After the suspect, Jeremy Moody, 30, was arrested on Wednesday, he acknowledged to the authorities that he had planned to kill again on Thursday. And he said that he had explained to Mr. Parker why he had been targeted.
"I'm not here to rob you," investigators say Mr. Moody told Mr. Parker. "I'm here to kill you because you're a child molester."
Mr. Parker had been convicted of sex offenses, but not child molesting.
Mr. Moody, a resident of nearby Lockhart, had long been a subject of monitoring by law enforcement officials, who followed his postings on social media Web sites before losing track of his online presence about a year ago.
Sheriff Taylor declined to elaborate on what specifically prompted his agency's concerns about Mr. Moody, who has a criminal record and the word "skinhead" tattooed across his neck.
"We've never thought of him as possibly being a serial killer, but he is someone who we have been watching for the last several years," Sheriff Taylor said. "We're in the South, and it isn't often you see people running around here with 'skinhead' tattooed under their neck."
Whether Mr. Moody has formal ties to white supremacist groups remains a focus of the investigation. Both the Parkers were white.
The authorities have not ruled out the possibility that Mr. Moody played a role in other crimes, and investigators have asked officials in neighboring counties to review whether their files contain any unsolved murders in which the victims were sex offenders.
Mr. Moody and his wife, Christine, have both been charged with two counts of murder in connection with the Jonesville killings, although her role appears to have been limited.
"I don't know if she originally knew that was what they were going there for, but she went in the house behind him while he had a gun in his hand," Sheriff Taylor said. "She knew they weren't there for lunch."
The murders have shaken Union County, the site of the well-known 1994 killings of two children by their mother, Susan Smith. It is otherwise a place largely without serious crime.
"You hate hearing anything like that," said Grady Carson, an employee at a hardware store where the Parkers shopped. "We've all done something in our past, and nobody wants someone to show up with a gun and play God."
Jack Levin, a criminologist who is co-director of the Brudnick Center on Violence and Conflict at Northeastern University, said it was likely that Mr. Moody possessed a common trait among killers.
"Serial killers often attempt to justify their killing sprees, and they usually do it by dehumanizing their victims," Dr. Levin said.
But the case in South Carolina is otherwise odd, he said.
"It's very unusual that someone would set out to kill large numbers of sex offenders."
Mr. Moody is not the first person accused of targeting sex offenders. As recently as last month, a California jury convicted a 36-year-old man of killing a neighbor who was a sex offender, and a Washington State man was sentenced in 2012 to life in prison for a pair of similar killings.