Parker made a personal-best five 3-pointers as the Commodores (10-14, 4-8 Southeastern Conference) won the first meeting between the schools since 1955.
Vanderbilt used a 20-6 run spanning the first and second halves to pull away from the Aggies (15-10, 5-7). Parker followed his final 3-pointer with a nifty reverse layup for the largest lead of the game, 46-33, with 8:37 left.
Texas A&M stormed back and closed the gap to 56-51 with 1:29 remaining. But Odom drained his career-high tying fourth 3-pointer to halt the rally.
Josh Henderson chipped in 11 for the Commodores, who had lost five of six.
A jumper by Henderson gave Vanderbilt what appeared to be a comfortable 56-45 lead with 3:18 to go. But the Aggies hung around. Kourtney Roberson tipped in a miss, Elston Turner followed with a jumper and a full-court press forced the Commodores into a turnover. Fabyon Harris made two free throws with 1:29 left to pull within five.
Vanderbilt, prone to relinquishing second-half leads this season, didn't panic. Odom got free in the left corner and made a game-clinching 3-pointer with 1 minute left.
It capped off a joyous day for the Commodores, who recognized three former players back in town. Festus Ezeli, John Jenkins and Jeffery Taylor -- three of the first 31 picks in last year's NBA draft -- were honored at mid-court early in the second half and received a standing ovation from the Memorial Gymnasium crowd. Just 11 months ago, the trio helped lead the Commodores to their third straight NCAA Tournament for the first time in program history.
The Aggies jumped ahead 7-0 early, making their first three shots. When Ray Turner threw down a two-handed dunk a little more than two minutes into the game, Vanderbilt coach Kevin Stallings quickly called a timeout and vented his frustration to his team. The Commodores responded with an 11-2 flurry that included two 3-pointers from Parker. Henderson's layup capped off the spurt and put Vanderbilt ahead, 13-11, for the first time with 13:37 remaining.
The teams exchanged leads in a slow-moving first half, but five quick points gave the Aggies their last lead of the half. Jordan Green soared in for an offensive rebound and putback. Ray Turner then muscled his way into the lane for a basket and made his free throw to complete the three-point play for a 21-19 lead with 2:57 to go until halftime.
Vanderbilt answered right back with a jumper by Odom, a forced turnover and a layup by Henderson.
Then with time winding down before the end of the half, the Commodores forced Elston Turner into a tough jumper that missed. Vanderbilt point guard Kyle Fuller drove the lane and made a pull-up jumper with 3 seconds left for a 25-21 halftime lead.
Odom kept the spurt going with a 3-pointer to open the second half. Henderson, Odom and Parker later scored in successive possessions to cap off the 20-6 run and take a 39-27 lead with 11:42 to go.
Parker, whose previous single-game high was two 3-pointers, hit his fifth in the corner with 9:05 to go. On the next possession, with the Aggies guarding the 3-point line, Parker drove the baseline, wiggled around Roberson and banked in a reverse layup with his right hand.
Texas Gov. Rick Perry sat in the first row behind the Aggies bench. The former Republican presidential candidate watched as his alma mater sputtered early and made just 9 of 23 shots (39.1 percent) in the first half.
adidas Men's Texas A&M Aggies Maroon Long Sleeve Shooter ShirtPrice: $47.99 Shop
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.