The 35-year-old Parker was demoted to the bench prior to Sunday's 94-86 loss to the Indiana Pacers after having started all but 13 times in the first 1,164 games of his career.
"I was like, 'Oh, no problem'," Parker said of being informed of the permanent move by coach Gregg Popovich. "Just like Manu (Ginobili), just like Pau (Gasol), that day's going to come. If Pop sees something that is good for the team, I'll try to do my best."
Popovich made the move with Ginobili in 2011. The 40-year-old has flourished as a backup, helping the Spurs win the 2014 NBA championship.
The Spurs took an impressive step toward demonstrating they're still a force to be reckoned with this season with a 114-102 home win over the Cleveland Cavaliers on Tuesday night.
Parker contributed 14 points and six assists in 18 minutes to the win, after having had 12 points and five assists in 20 minutes in the loss to Indiana.
New starter Dejounte Murray was a major player in the win over Cleveland, totaling 19 points, 10 rebounds and seven steals. He had eight points and seven assists against the Pacers.
Whether it's been as a starter or as a reserve, Parker has had a winning history against the Grizzlies. He's gone 43-12 in 55 matchups, averaging 15.1 points and 5.7 assists.
He's come off the bench twice in those 55 games, averaging 14.0 points and 7.5 assists in those contests.
He's started the last 26 times he's faced the Grizzlies, going 21-5 in those games.
Two of the latter wins came earlier this season. He averaged 9.0 points and 4.5 assists in starts during Spurs victories over Memphis in November and December.
The Spurs have hit a rare rough patch in the season, with Tuesday's win over Cleveland being their first in the last three games. They haven't recorded back-to-back wins since December.
Even the schedule has been unkind to the Spurs. They've had a one-day trip to Memphis placed between two- and five-game homestands.
Meanwhile, they catch the Grizzlies in their best form since October. Monday's 105-101 home win over Philadelphia was their fourth in the last five games.
Tyreke Evans, who has made the transition from backup to starter with Mike Conley out of the lineup, had 18 and Marc Gasol 19 against the 76ers. Evans has scored in double figures in 24 straight starts.
Evans had 10 fourth-quarter points among his 18 against the 76ers, who led the Grizzlies by 10 with 9:45 remaining.
Two-way signee Myke Henry also contributed big-time to the rally with a pair of fourth-quarter 3-pointers.
Henry, who scored his first NBA points on a 3-pointer earlier in the game, was signed out of the NBA G-League on Jan. 13.
"Life comes at you fast, and you, obviously, have to be ready for changes," said Henry, a silver medalist at the 2016 FIBA Three-on-Three World Championships in China. "I was ready. It's been a journey, and I'm just taking it day-by-day."