Parents listed on census are William H and Cynthia Bagley.
1900 census indicates she is an adopted daughter
Sources
United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MZZC-HDD : 14 July 2016), Gertrude G Bagley in household of William H Bagley, Watts Flats, Chautauqua, New York, United States; citing enumeration district ED 61, sheet 447A, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d.), roll 0816; FHL microfilm 1,254,816
United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MS6N-K17 : accessed 21 January 2017), Gertrude G Bagley in household of William H Bagley, Harmony Township (east part), Chautauqua, New York, United States; citing enumeration district (ED) 100, sheet 2B, family 51, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, 1972.); FHL microfilm 1,241,014
United States Census, 1910," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M54F-WBB : accessed 21 January 2017), Gertrude Willets in household of Arthur Willets, Harmony, Chautauqua, New York, United States; citing enumeration district (ED) ED 143, sheet 7B, family 171, NARA microfilm publication T624 (Washington D.C.: National Archives and Records Administration, 1982), roll 929; FHL microfilm 1,374,942.
United States Census, 1930," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:X4RB-XDR : accessed 21 January 2017), Gertrude G Willets in household of Arthur Willets, Cornwall, Orange, New York, United States; citing enumeration district (ED) ED 7, sheet 24A, line 8, family 78, NARA microfilm publication T626 (Washington D.C.: National Archives and Records Administration, 2002), roll 1632; FHL microfilm 2,341,366.
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