In the 1911 Census Roy’s birthplace is given as Parkgate; Yorkshire and the address is given as 76 Albert Road, Parkgate, Nr Rotherham.
A search of FreeBMD for all Rogers births within the registration district of Rotherham between 1908-1910 threw up this one:
Rodgers, Wroy; Mar Qtr 1910; District Rotherham; Vol 9c page 805
May be worth searching the medal rolls for that spelling.
I don’t know where that spelling of Roy originates, but it may have some religious source and I notice that in 1891 the Rodgers family are all involved with a local Chapel.
Going off the topic of the Navy and you may already have some of this.
Looks like his parents’ marriage was in Worksop (Nottingham/ Derbyshire border) (at least it was the only one that fits)
Rodgers, Wilfred; District Worksop; Sep Qtr 1901; Volume 7B, Page 54
Goodall, Gertrude; District Worksop; Sep Qtr 1901; Volume 7B, Page 54
Likely birth registration for Wilfred
Rodgers Wilfred; Jun Qtr 1876; District: Worksop; Vol 7b page 51
Possible but a less likely birth registration
Rodgers Wilfred Sharp; Sep Qtr 1875; District: Chesterfield; Vol 7b page 664
Note: until 1937 the Worksop Registration District included the parish of Clowne when it was transferred to Chesterfield Registration District.
Address: 9, Bridge Terrace, Worksop
RODGERS, Mary A; Head; Widow; age 59; born Pleasley, Nottinghamshire
RODGERS, Wilfrid; Son; Single; age 24; Grocer's Manager; born Clowne, Derbyshire
HUDSON, Helena; Boarder; Single; age 24; School Teacher; born Hunwick, Durham
GOODALL, Gertrude; Visitor; Single; age 21; Managing Dressmaker; born Thorpe Salvin, Yorkshire
RG13; Piece: 3124; Folio: 6; Page: 3
Registration District: Worksop; Civil Parish: Worksop
Address: 4, Coffee Tavern, Bridge Place, Worksop, Nottinghamshire
RODGERS, George; Head; Married; age 41; Coffee Tavern & Chapel Keeper; born Clown, Derbyshire
RODGERS, Mary Ann; Wife; Married; age 48; Assistant Coffee Tavern & Chapel Keeper; born Pleasley Hill, Nottinghamshire
RODGERS, Ada Ann Daughter Single16 Assistant Chapel Keeper Clown, Derbyshire
RODGERS, Wilfred Son14 Assistant Chapel Keeper Clown, Derbyshire
EKIN, Alice Servant15 General Servant Worksop, Nottinghamshire
RG 12, Piece: 2643, Folio: 111, Page: 6
Registration District: Worksop; Civil Parish: Worksop; County: Nottinghamshire