Answer: R.A. Dickey
R.A. Dickey made his MLB debut in 2001 for the Texas Rangers at age 26. He did not play in 2002 or 2007 but was in the league every other year until 2017.
Dickey was a veteran that was brought in by the Braves organization to be a clubhouse leader. A lot of Braves fans will remember Dickey as unsuccessful in his time in Atlanta but he actually was not that bad in 190 innings pitched.