Payton sat out all of the 2012 season due in a suspension brought on by evidence the Saints had paid defensive players to purposely injure members of the opposition.
New Orleans went 7-9 without their head coach last year.
Quarterback Drew Brees went 7-of-10 for 65 yards while playing two series. He drove the Saints 74 yards to a field goal in the second period.
Luke McCown came on to complete 18-of-28 for 216 yards and he threw the two touchdown passes to Parker -- both of them for 2 yards.
Andy Reid made his coaching debut for Kansas City after guiding the Philadelphia Eagles for 14 seasons.
Alex Smith, who came from San Francisco to quarterback Kansas City this year, went 7-for-8 for 68 yards.
New Orleans overcame a 13-point deficit.