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 2966, the Parker puppet is used as a female monster (Camille Bonora) who keeps calling Telly Monster, who's filling in for the Grand High Triangle Lover, and asking him which triangle related bath materials she should use (and then not hearing his answers due to having a shower cap over her ears).