Hi Amanda,
You're not mentioning at all, the middle child of William & Mary , Mary Bruce WALKER born 1868, even though she's there on Family search and there in the 1871 Census? She may hold keys as to what became of her parents and why you (we) can't find them in 1881.
The 1886 marriage Maureen found, in fuller detail:
Mary Bruce WALKER
M 23 Oct 1886, at age 18, at Birstall St Peter to:
Wellington SCHOLFIELD, age 24
Both of Heckmondwicke
Brides: Father William, Mariner
Witness: Walliam WALKER (the other witness was a Scholefield)
That marriage is I'd say, pretty much certain to be 'your' Mary (later census show she was born Aberdeen). It's in the right area of Yorkshire (where her 2 siblings are marrying in). In which case, the witness at Mary's wedding, William WALKER, is likely to be her father. He has signed his name in large, bold, writing though not terribly 'literate' looking. If that signature could be compared to that which may be on the marriage of William WALKER & Mary BRUCE, and there is a match....
The Census Maureen found, in full detail....shows the other 2 children in the household
Census In 1881: Gawber, in Barugh, Yorkshire (Barnsley district)
RG11 /Piece 4604 /Folio: 167 / Pg29
Head: Wm WALKER, 42, Rope Repairer, b Scotland
Wife: Eliz. WALKER 40, b Scotland
Dau: Mary WALKER 13, Dom Serv, b Scotland
Son: Andrew WALKER 11, b Scotland
Son: John WALKER 8, b Scotland
Then there is this:
In 1891: ?Answorth Road, Ilkeston, Derbyshire
RG12 /Piece 2664 / Folio 104A / Pg38
Head: William WALKER 54, married, (Wire) Rope Worker, b Aberdeen, Scotland
Step-Son: John DAVIDSON 15, Apprentice Engine Fitter, b Stranra, Scotland
Visitor: Alexander SOUTTER 51, married, Butcher, b Aberdeen, Scotland
Census In 1891,
Wellington SCHOEFIELD & wife Mary are in hamlet of Norristhorpe Lane, Liversedge, Yorkshire, with their 1 yr old daughter, Mary. Wife Mary is 23, born (abt 1868), Aberdeen Scotland
RG12/ Piece 3726 / Folio 32 / Pg20
Liversedge is in the Dewsbury Registration District.
Cheers
AMBLY