That same year, Parker was noticed by critics when she appeared in a movie called "Longtime Companion". It was one of the first movies about AIDS. After that, she acted in 1991's "Grand Canyon", which also starred Mary McDonnell, Danny Glover, Alfre Woodard and Kevin Kline. Parker's next film was "Fried Green Tomatoes", with Jessica Tandy, Mary Stuart Masterson, Kathy Bates and Cicely Tyson.