Parker began his professional career in 2006 as a first baseman, third baseman, and catcher, and did not start pitching until 2007, splitting the season between the AZL Cubs and Boise Hawks, with a combined record of 2–0 and a 2.39 ERA. He was an AZL post-season All Star in 2006. In 2008, Parker played for three teams—the Peoria Chiefs, the Daytona Cubs, and the Iowa Cubs —going 4–2 with a 2.13 ERA and 12 saves in 45 games combined. In 71 innings, he allowed only 50 hits while striking out 75 batters. He split the 2009 season between the Tennessee Smokies and Iowa Cubs, going a combined 2–3 with a 2.70 ERA, saving 25 games in 55 appearances. His 22 saves with Iowa were 5th-most in the Pacific Coast League. In 63 innings, Parker allowed 44 hits and struck out 77 batters. He then played for Mesa in the Arizona Fall League, and was named a Rising Star. Parker began the 2010 season with the Iowa Cubs, and split the year between Iowa and Tennessee, going a combined 1–5 with a 4.21 ERA. In 2011, he went 4–5 with a 3.23 ERA in 53 games split between Iowa and Tennessee. The Cubs promoted him to MLB on May 17, 2012. He pitched only six innings with the Cubs in 2012. In 2013 with the Cubs he was 1–2 with one save and a 2.72 ERA, in 49 games, as he struck out 55 batters in 46 innings. In 2014 with the Cubs he was 1–1 with a 5.14 ERA, in 18 games, as he struck out 24 batters in 21 innings. With Iowa, he was 0–1 with a league-leading 25 saves and a 1.77 ERA in 35 games. He was a Pacific Coast League mid-season and post-season All Star in 2014. In 2015 he did not pitch in the major leagues, and was limited to three games at Triple-A Iowa, as he had season-ending surgery to remove loose bodies in his right elbow. Parker was released from the Cubs in May 2015.
Seattle Mariners (2016)
On December 17, Parker signed a minor league deal with the Seattle Mariners. The Mariners promoted Parker to the major leagues on August 4, 2016, and he pitched only one scoreless inning for them in 2016. In 2016 he was again a Pacific Coast League mid-season All Star, playing for Tacoma.
New York Yankees (2016)
On August 9, 2016, the New York Yankees claimed Parker off of waivers. With the Yankees, he was 1–0 with one save and a 4.96 ERA in 16 games. After the 2016 season, the Los Angeles Angels claimed Parker from the Yankees off of waivers. On November 23, 2016, Parker was claimed off waivers by the Milwaukee Brewers. The Brewers designated him for assignment in December, and he was claimed by the Angels on December 23.
Los Angeles Angels (2017–18)
The Angels named Parker to their Opening Day25-man roster for the 2017 season. In his first season as an Angel, Parker enjoyed career bests in every statistical category. He was 3–3 with 8 saves and a 2.54 ERA, with 86 strikeouts in 67 innings, and pitched in 71 games. His WHIP of 0.83 tied for third-best among American League relievers. Parker signed a $1.8 million contract with the Angels for the 2018 season. Despite pitching to a 2–1 record with a 3.26 ERA in 67 games while also collecting 14 saves, striking out 70 batters in 66 innings, and finishing 41 games, he was non-tendered by the Angels in the offseason.
On July 30, 2019, the Philadelphia Phillies signed Parker to a major league contract. For the Phillies in 2019, he was 2-1 with a 5.04 ERA in 23 games over 25.0 innings in which he struck out 31 batters. Parker re-signed with the Phillies organization on a minor league deal on February 5, 2020.
Personal life
Parker and his wife, Jordan, have two sons: Jude and Richard Tyce.