Parker was born in 1964 in Fort Jackson, South Carolina. Her mother was Swedish and her father was a judge and served in the United States Army. They lived in Thailand where she attended the International School Bangkok. Parker studied drama at the North Carolina School of the Arts. Her first job in television was a small part on the soap opera "Ryan's Hope" in 1975. In the late 1980s, she moved to New York, where she got a job measuring feet at ECCO (a shoe shop). After a few small roles, she had her first part in a Broadway play in 1990. The play was called "Prelude to a Kiss" and she played the main character called Rita. She won the Clarence Derwent Award for this play and was nominated for a Tony Award. Parker dated her co-star Timothy Hutton for a short time. 