Warner's parents were John J Bishop and Eliza Bell Bishop, both from New York, USA
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Columbia University College of Physicians and Surgeons Obituary Database has the following obituary for him:

F. Warner Bishop
Record: 3958
Death Date: March 23 1947
Graduating Year: P&S 1912

Secretary of his class at P&S, died after a long illness at St. Luke's Hospital, New York City, where he was senior attending physician and director of the out-patient department. In addition to his work at St. Luke's, whose staff he joined in 1915, Dr. Bishop was on the staff of the Sloane Hospital for Women, and was an instructor in physiology and assistant in clinical medicine at Columbia. Beside holding the secretaryship of the Class of 1912, Dr. Bishop was a former president of the Society of the Alumni of St. Luke's Hospital, of the New York Clinical Society and of the Hospital Graduates Club.

We can see Darrette in 1900 living at Bayonne Ward 1, Hudson, New Jersey
Age: 9
Birth Date: Jun 1890
Birthplace: New Jersey
Race: White
Ethnicity: American
Gender: Female
Relationship to Head of House: Daughter
Father's Name: Edward
Father's Birthplace: Wales
Mother's Name: Emily
Mother's Birthplace: England
Marital Status: Single
Residence : Bayonne City, Hudson, New Jersey

Living in the household were:
Edward Hoare 59, 9/40 Wales - civil engineer
Emily Hoare 37, 4/63 ENG
Edna Hoare 18, 8/81 ILL
Darrette Hoare 9, 6/90 NJ
Lillian Hoare 7, 2/93 ENG
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In 1910 we see her living at Elizabeth Ward 10, Union, New Jersey
Name: Dorette Hoare
[Dorette Haare]
Age in 1910: 19
Estimated Birth Year: abt 1891
Birthplace: New Jersey
Relation to Head of House: Daughter
Father's Name: Edward
Father's Birth Place: Wales
Mother's Birth Place: England
Marital Status: Single
Race: White
Gender: Female

Living in the household were:
Edward Hoare 69, ENG/Wales - civil engineer, widowed
Edna Hoare 28, ILL - nurse
Dorette Hoare 19, NJ
Lillian Hoare 19 [17] NJ
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The again in 1920 she was living at Manhattan Assembly District 9, New York, New York
Name: Dorette Bishop [aka = Hoare]
Home in 1920: Manhattan Assembly District 9, New York, New York
Age: 29 years
Estimated Birth Year: abt 1891
Birthplace: New Jersey
Relation to Head of House: Wife
Spouse's Name: F Warner
Father's Birth Place: England
Mother's Birth Place: England
Marital Status: Married
Race: White
Sex: Female
Able to read: Yes
Able to Write: Yes

Living in the household were:
F Warner Bishop 32, NY - physician
Dorette Bishop 29, NJ
Dorathy Bishop 2 4/12, NY
John J Bishop 56, NY - father
Eliza Bell Bishop 55, NY - mother