I realize I'm ludicrously late for this, but the very first parody film was The Little Train Robbery, made in 1905. It was a parody of The Great Train Robbery, made in 1903, which had the same director.
a musical, literary, or other composition that mimics the style of another composer, author, etc, in a humorous or satirical way.
I would nominate the Laurel and Hardy movies as parodies of the originals. Movies were still in their infancy when Stan Laurel and Oliver Hardy created shorts and full length feature films in the 1920s with monsters to comedic effect. I'm sure I'd do better if I knew my early 20th century film, but this is my first recognizeable case in point (from IMDB):
Dr. Pyckle and Mr. Pryde in 1925, parodies Dr. Jekyll and Mr. Hyde with John Barrymore (Grandfather of Drew Barrymore) in the lead roles of Jekyll and Hyde in 1920.
Abbott and Costello took up the idea in the 40's with Frankenstein, The Wolfman, and Abbott and Costello Meet Dr. Jekyll and Mr. Hyde in 1953.
Intolerance intercuts four parallel story lines each separated by several centuries and demonstrates mankind’s persistent intolerance throughout the ages.
Three Ages has three plots in three different historical periods. It tells about the love of a man to a woman. Some background can be found in a TCM article.
According to Parody As Film Genre: "Never Give a Saga an Even Break" (1999) by by Wes D. Gehring:
Unlike other American film comedy genres such as the 1930s Depression birth of screwball comedy, or the front and center emergence of dark comedy in the 1960s, parody has been a mainstream part of American film comedy since the beginning. It was, in fact, a pivotal ingredient in the works of Mack Sennett, America's film comedy father. This comedy pioneer was at his best when spoofing the melodramatic adventure films of his mentor, D. W. Griffith. For instance, Sennett's Teddy at the Throttle (1917) is a delightful takeoff on Griffith's propensity for the last-minute rescue, such as the close of his celebrated but still controversial Birth of a Nation (1915).
... Film comedy historian Gerald Mast reminds us that Sennett's Help! Help! (1912) is a "specific parody" of Griffith's Lonely Villa (1909), which Sennett had written.