Parcells' agent, Jimmy Sexton, declined comment to The Associated Press. Saints spokesman Greg Bensel said the team had "nothing to report" on its search for an interim coach.
The Saints now will look to hire from within the organization. By promoting an assistant on the team, the Saints will not have to interview a minority candidate in accordance with the NFL's Rooney Rule.
The Saints' coaching search is being led by general manager Mickey Loomis and owner Tom Benson, who will make the final decision on who will replace Payton next season, a source with knowledge of the situation told NFL.com's Steve Wyche. Payton is serving in an advisory capacity in discussions about what the team's course of action will be, Wyche reports.
Assistant head coach Joe Vitt, who has been suspended for the first six games of the upcoming season, and defensive coordinator Steve Spagnuolo are among the internal candidates who could fill in for Payton.
On Monday, NFL Commissioner Roger Goodell upheld Payton's suspension for the 2012 season, along with an eight-game suspension for Loomis and a six-game ban for Vitt.
Payton and Parcells spoke in March about a possible role with the Saints for the retired coach. Parcells, who's a finalist for this year's Pro Football Hall of Fame class, said he felt "an obligation" to assist Payton, but he remained coy about if he'd fill in for his longtime friend.
Payton's season-long suspension without pay will begin April 16. NFL spokesman Greg Aiello told NFL Network's Albert Breer that Payton will not "be involved in any coaching activities, he can't be at the facility or with the team at games" once his suspension begins.
Follow Jason La Canfora on Twitter @jasonlacanfora.
The Associated Press contributed to this report.