Yester-day I watched My Fair Lady for the first time in years (and I think for only the second time ever). I think I was dimly aware that The Monkees episode "The Chaperone" spoofed some elements from the movie (the trivia on The Monkees DVD briefly mentions as much), but since I'd just watched the movie, I also watched "The Chaperone" to see what specific elements are borrowed or parodied.
After Davy has the idea to employ Mrs. Weefers as the chaperone for the Monkees' party, the show cuts to a scene where he and Peter are dressed to resemble Henry Higgins and Colonel Pickering, and Davy tries to teach Mrs. Weefers the correct pronunciation of the sentence "The dance in France is mainly in the stance," which is similar to a sentence whose correct pronunciation Henry Higgins tries to teach Eliza Doolittle: "The rain in Spain stays mainly in the plain."
Davy then says, "I tell you, Peter, give me six months, and I could pass her off as a duchess at an embassy ball." This is nearly verbatim what Higgins claims: "In six months, I could pass her off as a duchess at an embassy ball."
In a later scene, Peter tries to teach Mrs. Weefers to speak clearly. He puts marbles in his mouth, and - increasingly impeded - says, "The idea is to put as many of these marbles into your mouth as you can and still...." In My Fair Lady, Higgins puts six marbles in Doolittle's mouth, with the aim to teach her to speak clearly: "I want you to enunciate every word just as if the marbles were not in your mouth."
Next, Davy holds a candelabrum before him as he delivers the line "Huntington Hartford hates pickled herrings." The reason for the candles isn't explained in the show, but in My Fair Lady, Higgins has Doolittle sit in front of a similar device and tells her, "Every time you pronounce the letter H correctly, the flame will waver, and every time you drop your H, the flame will remain stationary." To demonstrate this, he speaks into the device: "In Hartford, Hereford, and Hampshire, hurricanes hardly ever happen." Davy's sentence - like Higgins' - is formed almost entirely of words that begin with H in order to teach correct pronunciation.