Pre-1900
Dexter Park dates back to the mid 1800's, when Hiram Woodruff owned the land next to Snedicker's Roadhouse. Hiram was an early sports superstar of his time; he raced the trotters at Union Course, L.I. Dexter Park is named after the famous racehorse Dexter. The grounds housed both a hotel and training ground.

1900-1930
Dexter Park becomes a significant sports location. The Brooklyn Royal Giants, a Negro League team, play there from 1905 to 1913 and 1923-1927. The Brooklyn Bushwicks, a semi-pro team, make it their home after their facility at Wallace Grounds burns down in 1917. Max Rosner, owner of the Brooklyn Bushwicks, buys Dexter Park in 1922 and builds a steel and concrete park with a capacity of 6,000. Soccer Championships are played there in 1929 , as well as college football games and boxing matches .

1930-1950
In 1930 Dexter becomes the first park to have permanent lights installed for night games. When major league baseball began playing the All-Star game in 1933, local pros who had the days off would play against the Bushwicks. Max Rosner, who owned the Bushwicks and the park, would get famous names to barnstorm at the park. Babe Ruth and Lou Gehrig played for the Bay Parkways in a game there in 1935. During the Second World War, pro players on leave would make a few bucks playing for the Bushwicks under fake names. Dodger Gene Hermanski played for the Bushwicks as "George Walsh". The Bushwicks were often considered an unofficial minor league team for the Dodgers because so many players moved back and forth, particularly during wartime. The park was popular with the large number of brewery workers in the area, who get get to the park cheaply and easily via the BMT line.

Post 1950

The advent of television spells doom for the park as an entertainment site. In 1951 a midget racecar track is installed around the outside perimeter of the ball field. Attendance continues to decline and in 1955 the 10 acres of grounds are sold to 213 Dexter Corp. for $405,000. In 1956 the ballpark is torn down and the land is used for housing.

Credit to David Dyte's Webpage from which I have borrowed liberally with permission. The link can be found here: http://www.covehurst.net/ddyte/brooklyn/dexter.html.
Thanks to Joe Scheibeler for directing me there.