The Commodores (15-11, 7-7 SEC) made 10 of 18 3-pointers in the second half after hitting just one before halftime.
The Tigers (12-13, 4-10) have lost four of five games and had a late scoring drought of 4 minutes, 28 seconds.
Parker led Vandy's 3-point barrage with three in a row during an 18-4 run early in the second half. He was 5 of 9 from 3-point range.
Damian Jones scored 14 points and Luke Kornet added 10.
K.T. Harrell and Chris Denson both scored 13 points for Auburn. But that was 14 shy of the combined average for the nation's No. 2 scoring duo.
Harrell was just 4-of-15 shooting and Denson made 3 of 15 shots.
Vanderbilt, which had lost three of four games, trailed 29-13 with less than seven minutes remaining in the first half before pulling to within seven by halftime.
The Commodores were 23-of-52 shooting (44.2 percent) in addition to clamping down on the Tigers' shooters. Auburn was 21 of 62 (33.9 percent) and made just 4 of 24 3-pointers (16.7 percent).
Kyle Fuller's 3-pointer with 2:28 left gave Vanderbilt its biggest lead, 66-57.
Auburn didn't end its drought until Allen Payne's jumper in the final minute.
Parker hit three consecutive 3-pointers during an 18-4 run early in the second half to give the Commodores their first lead, going ahead 43-38. Vandy's six baskets during that stretch were all 3s.
Then Tahj Shamsid-Deen scored on three consecutive possessions for Auburn and tied it 45-45 with a 3-pointer with 12:33 left. Those were his only seven points.
Odom, who came in averaging a team-high 14.5 points, didn't score until midway through the second half. He followed that up with a pair of 3-pointers and finished with eight points.
The Commodores, down to seven scholarship players, had to turn to the walk-ons with several starters having foul trouble.
Nathan Watkins had seven points and Carter Josephs dished out nine assists.
The comeback win came after the Commodores had fallen short after erasing most of an eight-point deficit over the final two-plus minutes in a 67-64 road loss to Missouri on Wednesday.
Parker, walk-ons pace Vandy past Auburn
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