In the 1861 Census for Calverley with Farsley, on the Enumeration District for the Farsley section of Stanningley, the enumerator took care to differentiate which part of Stanningley a person was born, but this is a unique example.
For registration purposes, a birth, marriage or death that took place in either the Calverley with Farsley (green) or Pudsey (blue) sections would show as being registered in Bradford (except for the years 1892 to 1898 when both Calverley and Pudsey came under North Bierley). The Bramley section of Stanningley (yellow) was in the district of Hunslet until 1862, then in Kirkstall until 1864, then Bramley until 1925, though at Leeds Register Office the entries are almost always shown as Kirkstall district.
Someone born in Stanningley could show up in any of these district registers, depending when they were born and which side of the road they lived on. Thankfully, all records from all these sub-districts are all together and can be found at Leeds Register Office, not Bradford. There is an exception. A few non-conformist churches in Farsley and Pudsey, notably Olivet Chapel on Bradford Road in the Farsley section of Stanningley, who had to have the registrar present for their marriages, came under North Bierley for longer than the officially prescribed period (1892-1898). Olivet’s marriages were under North Bierley until 1924. These marriage records should be at Bradford.