Answer by taking a quote from the following article:

Clemens was born in Dayton, Ohio, the fifth child of Bill and Bess (Lee) Clemens. He is of German descent, his great-grandfather Joseph Clemens having immigrated in the 1880s. Clemens's parents separated when he was an infant. His mother soon married Woody Booher, whom Clemens considers his father.

In 1999, while many of his performances and milestones were yet to come, he ranked number 53 on The Sporting News' list of the 100 Greatest Baseball Players, and was elected by the fans to the Major League Baseball All-Century Team. In 2005, the updated Sporting News list moved Clemens up to #15.  By the end of the 2005 season, Clemens had won seven Cy Young Awards (he won the AL award in 1986, 1987, 1991, 1997, 1998, and 2001, and the National League award in 2004), an MVP and two pitching triple crowns. With his 2004 win, he joined Gaylord Perry, Randy Johnson, and Pedro Martinez as the only pitchers to win it in both leagues and became the oldest pitcher to ever win the Cy Young. He has also won the Sporting News Pitcher of the Year Award five times, was named an All-Star 11 times, and won the All-Star MVP in 1986.  In October 2006, Clemens was named to Sports Illustrated's "all-time" team.  On August 18, 2007, Clemens got his 1,000th strikeout as a Yankee. He is only the ninth player in major league history to record 1,000 or more strikeouts with two different teams. Clemens has recorded a total of 2,590 strikeouts as a member of the Red Sox and 1,014 strikeouts as a Yankee. Of his nearly quarter century in the Major Leagues, 13 years have been spent with the Red Sox and 6 with the New York Yankees.  In January 2013, in his first year of eligibility, Clemens received only 37.6% of the votes cast and was denied entry into the Hall of Fame, falling short of the 75% required for induction. With the inductions of Greg Maddux and Tom Glavine the following year and Randy Johnson in 2015, Clemens is currently the only eligible member of the 300 win club not to be inducted into the Hall.

Were those the only years he won the Cy Young?
and the National League award in 2004),