Parker set a career-high and Odom scored 16 points in the first half for the Commodores (14-16, 8-10 Southeastern Conference), who fended off a late charge.
Vanderbilt led by 12 in the second half but South Carolina (14-17, 4-14) pulled within two, 56-54, on Lakeem Jackson's layup with 3:45 left. Josh Henderson answered with a layup and Parker knocked down a 3-pointer.
The Gamecocks cut the deficit to four twice in the final two minutes but Vanderbilt made five straight free throws to seal its fifth straight victory in the series.
Vanderbilt leading scorer Kedren Johnson scored 10 of his 12 points -- all coming in the second half -- from the free-throw line.
With a 36-25 halftime lead, the Commodores missed eight of their first nine shots after the break. Meanwhile, South Carolina heated up.
Brenton Williams made consecutive 3-pointers and a stepback jumper. Brian Steele curled around a screen and uncorked a 3-pointer to pull South Carolina within three, 43-40, with 12:28 left. Vanderbilt's Sheldon Jeter answered with a 3 in the corner to spark a 7-2 run and stretch the lead to eight points.
South Carolina hung around, though, and closed the gap to 56-54 on a layup by Lakeem Jackson. But his unconventional one-handed free throw missed and failed to complete the three-point play.
Then Henderson drove in for a layup and Parker banked in his 3-pointer for a 61-54 lead with 2:23 left. Williams' third 3-pointer off an offensive rebound chopped the deficit to 66-62 with 35 seconds remaining. Vanderbilt silenced the comeback by making eight of 10 free throws in the last 51 seconds. The Commodores, who two months ago were the second-worst free-throw shooting team in the country, made 23 of 28 from the foul line.
Odom rattled off 12 straight points in a three-minute stretch at the end of the first half to help the Commodores build a 12-point lead.
Despite shooting 26.7 percent (8-of-30) in the first half, the Gamecocks had trimmed the lead to four, 22-18, with 4:32 left in the half. Parker drove the lane and kicked it out to Rod Odom for an open 3. After a Shelby Moats steal, Parker again found Odom, this time in the opposite corner, for another 3-pointer. He stayed hot with a jumper and two free throws for a 32-20 lead. Cutting to the basket, he banked in a layup to end his scoring streak with 70 seconds before the halftime.
An alley-oop slam by R.J. Slawson chopped the South Carolina deficit to single digits momentarily. But Kyle Fuller's driving layup rolled in as time expired to give Vanderbilt an 11-point halftime lead.
Vanderbilt does not have any seniors on the roster but lone senior student manager Daniel Marks was honored prior to the last home game of the season. Coach Kevin Stallings presented him with a new mop to wipe up the Memorial Gymnasium floor one last time.
The Commodores head into next week's SEC tournament as the No. 10 seed and winners of six of nine. Vanderbilt won the SEC tournament championship last year for the first time since 1951. But all five starters and top six scorers from last year's team are gone.
The Gamecocks have lost four of five and will be either the No. 12 or No. 13 seed in the SEC Tournament, which will be held in downtown Nashville.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.