From 1994 to 1997, Parque attended UCLA and led the Bruins to the College World Series in 1997. Parque earned second-team Smith Super Team honors in his sophomore season in 1996. In his junior year, Parque was voted first-team All-American by Baseball America, first-team All-Pac-10 Conference, second-team by the Sporting News, second-team by the American Baseball Coaches Association, and third-team by Collegiate Baseball. Parque is one of the most decorated pitchers in UCLA Baseball history. He currently ranks second in career games started with 50, second in career total innings pitched with 334⅔ innings, second in career strikeouts with 319, third in career pitching wins with 25, and seventh in career complete games with 10. In terms of single season pitching records for the Bruins, Parque ranks third in wins with 13 in 1997, ninth for games started with 19 in 1997, ninth for innings pitched with 125⅔ in 1996, fourth in strikeouts with 119 in 1997, and fifth in strikeouts with 116 in 1996.
In the 1997 supplemental draft he was drafted by the Chicago White Sox in the 1st round. Parque spent his first five seasons with the White Sox. He enjoyed his best season in 2000, going 13–6 with a 4.28 ERA before being beset by arm problems (torn labrum).
On June 24, 2004, he announced his retirement after playing seven seasons of professional baseball due to his recurring arm injury from 2000.
After being out of baseball for three years, Parque announced his willingness to return to the game of baseball. The Chicago Tribune reported that he threw his fastball in the range of 90 mph. On February 2, 2007, he signed a minor league contract with the Seattle Mariners. He was released by Seattle on May 31, 2007. He has since been linked to steroids in December 2007, though he denied the account in the Seattle Times. In a July 23, 2009 article in the Chicago Sun-Times, Parque admitted using human growth hormone while rehabbing from a shoulder injury in 2003.