You can trace this family on familysearch.org. It appears that Grover Cleveland's Furlow's father was Robert Hill Furlow and his mother was Vriginia E Goodwin.The family is found in census data as well as death records.
As far as spelling of the name - census takers often wrote down what 'they heard' and it was not necessarily spelled out for them. So you get varied spellings. Same with children's names, which may have been a nickname and not a birth name given at the time of the census. On one census mother is Vaginia E rather than Virginia E.
https://familysearch.org/pal:/MM9.3.1/TH-266-11606-133990-63?cc=1320964
Texas Deaths, 1890-1976 004168244 Image 2150
Name: Grover Cleveland Furlow
Death Date: 21 Jun 1976
Death Place: Karnes City, Karnes, Texas
Gender: Male
Race: White
Death Age: 91 years
Birth Date: July 17, 1884
Birthplace: Houston County, TX
Marital Status: Never Married
Father's Name: Robert Hill Furlow
Mother's Name: Virginia Elizabeth Goodwin
Burial Place: Grove Hill Cem., Dallas, Dallas, Texas
Burial Date: 22 Jun 1976
Film Number: 2243939
Digital Film Number: 4168244
Image Number: 2150
Reference Number: 54482
Census data:
https://familysearch.org/pal:/MM9.1.1/MFNL-X7C
United States Census, 1880 Precinct 4, Houston, Texas)
Name: Robt. H Furlow
Residence: Precinct 4, Houston, Texas
Birthdate: 1842
Birthplace: Mississippi, United States
Relationship to Head: Self
Spouse's Name: Virginia Furlow
Spouse's Birthplace: Alabama, United States
Father's Birthplace: Georgia, United States
Mother's Birthplace: Tennessee, United States
Race or Color (Expanded): White
Ethnicity (Standardized): American
Martial Status: Married
Age (Expanded): 38 years
Occupation: Farmer
NARA Film Number: T9-1312
Page: 360
Page Character: C
Entry Number: 3487
Film number: 1255312
Household Gender/Age
Robt. H Furlow M/38
Spouse Virginia Furlow F/30
Child Mcanally Furlow M/9
Child Thomas R. Furlow M/7
Child Maggie M. Furlow F/5
Child Nettie R. Furlow F/3 (Nonnie? in 1900)
Child Robt. H. Furlow M/1 (Hill? in 1900)
Another census. "Vaginia E is listed as widow' so Robert H must have died sometime between 1893 and 1900 based on birth of daughter Ruby in 1894.
https://familysearch.org/pal:/MM9.3.1/TH-266-11077-146751-78?cc=1325221
United States Census, 1900 Texas Houston 0025 Justice Precinct 8
Image 29
Name: Vagima E Furlow
Residence: Justice Precinct 8, Houston, Texas
Birth Date: Apr 1850
Birthplace: Alabama
Relationship to Head of Household: Self
Father's Birthplace: Alabama
Mother's Birthplace: Alabama
Race or Color (expanded): White
Head-of-household Name: Vagima E Furlow
Gender: Female
Marital Status: Widowed
Mother How Many Children: 6
Number Living Children: 6
Enumeration District: 0025
Page: 15
Sheet Letter: A
Family Number: 300
Reference Number: 36
Film Number: 1241647
Image Number: 00281
Vagima E Furlow F/50 b Apr 1850 AL
Child Maggie Furlow F/23 b Feb 1877 TX
Child Hill Furlow M/21 b. May 1879 TX
Child Nonnie Furlow F/10 b. Oct 1889
Child Grover Furlow M/15 b. Jul 1884
Child Zula Furlow F/13 b Jan 1887
Child Ruby Furlow F/3 b Aug 1894
Possibly father:
https://familysearch.org/pal:/MM9.1.1/M4LD-SGP
United States Census, 1850
Name: Wm T C Furlow
Event: Census
Event Date: 1850
Event Place: Lauderdale county, Lauderdale, Mississippi, United States
Age: 33
Birthplace: Georgia
Estimated Birth Year: 1817
Dwelling House Number: 63
Family Number: 64
Line Number: 22
NARA Publication Number: M432
NARA Roll Number: 375
Film Number: 443587
Digital Folder Number: 004200046
Image Number: 00214
Wm T C Furlow M/33
Kizziah Furlow F/34
Albert M Furlow M/9
Robt H Furlow M/8
Charles T Furlow M/6
Francis E Furlow F/1
James M Furlow M/29
Deaths of Siblings:
https://familysearch.org/pal:/MM9.3.1/TH-266-11852-80270-25?cc=1320964
Name: Maggie May Bounds
Death Date: 20 Apr 1924
Death Place: Port Arthur, Jefferson, Texas
Father's Name: Robert H. Furlow
Father's Birthplace: Mississippi
Mother's Name: Virginia Goodwin
https://familysearch.org/pal:/MM9.3.1/TH-266-11852-101385-22?cc=1320964
Name: T.K. Furlaw
Death Date: 13 Apr 1924
https://familysearch.org/pal:/MM9.3.1/TH-267-13030-45889-49?cc=1320964
Name: Nattie Rebecca Green
Death Date: 26 Jun 1961
Death Place: Kenedy, Karnes, Texas
Father's Name: R.H. Furlow
Mother's Name: Virginia E. Goodwin
https://familysearch.org/pal:/MM9.3.1/TH-266-12050-27910-2?cc=1320964
Name: Mack N. Furlow
Death Date: 30 Oct 1917
Death Place: Dallas, Dallas, Texas
Father's Name: Robt. H. Furlow
Father's Birthplace: Miss.
