Parker is a blue monster with a green nose and a tuft of white feathers on his head. He appeared during the late 1980s as one of Elmo's friends, most notably in Linda Ronstadt's song "La Charreada" (played by Jerry Nelson).
He has also appeared in various other sketches on the show, sometimes just as an extra. One notable example was on the game show "What's My Job?", where he was played by David Rudman.
In episode 3243 the Parker puppet was used as a female monster who keeps calling Telly Monster, who is acting as the Grand High Triangle Lover of the Day, and asking him which triangle related bath materials he should use (and then not hearing his answers due to having a shower cap over her ears), and driving him absolutely up a wall.