It appears by the age of Harry G Walker, that they may have been in Panama around 1908-1910? I don't know where you would look for those records, but maybe someone else can help. You might try passenger lists between 1910-1920 to see if you can find their re-entry, and if a father is listed.
World War I Draft Registration
https://familysearch.org/pal:/MM9.3.1/TH-1951-26614-10983-73?cc=1968530
Name: Reginald George Walker
Residence: RFD Box # 79 Sharpsburg
Draft Registration Sept 12, 1918 Sharpsburg, Allegheny Co., no 16, PA
White Male Birth 10 Dec 1898 Birthplace: US
Pipefitter, Atlantic Refining Co.,
Nearest Relative: Adina Walker RFD Box #79, Sharpsburg
U S Census, 1920 O'Hara, Allegheny, PA
Citing this Record: "United States Census, 1920," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M6BK-XXW
Head Adina P Walker F 46 RI/Eng/Scot Widow
Son Reginald G Walker M 21 NY/Scot/RI
Son Harold G Walker M 11 Panama/Scot/RI
