It looks as though the parents married at St Thomas Brightside, Sheffield in 1886. His name was entered at the time as Thomas Henry Rodgers, but according to a FreeBMD postem it was amended in the parish register to Thomas Henry Oldfield years later, on 4 August 1930.
According to another postem the bride’s name was amended at the same time, from Kate Houlston to Catherine Holton.
Original GRO reference: Sep qtr 1886 Sheffield 9c 405
Amended GRO reference: Sep qtr 1886 Sheffield 9c 404b