Two brief answers to your inquiries:
Father, Alexander McBurney, died 23 SEP 1896. Mother, Elizabeth Noble, died sometime between SEP 1896 and MAR 1918.
In more detail:
Circa 1834, Alexander McBurney was born in Garnkirk, parish of Cadder, Lanark. His parents were William McBurney and Mary McAllister.
20 OCT 1851, Alexander McBurney married Sophia McNeilly in Dunfermline, Fife. They had three known children.
1871 census, Alexander and Sophia were living in Upholland, Lancashire, England. Alexander’s occupation was brick maker.
Sometime after the 1871 census, Alexander and Sophia either separated or were divorced.
01 MAY 1861, Elizabeth Noble was born in Tranent, East Lothian. Her parents were William Noble and Elizabeth Davidson.
1881 census, Elizabeth Noble was in Carrington, Midlothian, living with her mother and siblings. Her occupation was brickwork labourer, rolling the clod.
30 JUL 1883, daughter Elizabeth McBurney (“McBurnie”) was born to Alexander McBurney and Elizabeth Noble in parish of Dalry, County of Ayr.
28 NOV 1887, son Alexander McBurney (“McBurnie”) was born to Alexander McBurney and Elizabeth Noble in parish of Markinch, County of Fife.
16 FEB 1889, Alexander McBurney and Elizabeth Noble were married in parish of Cockpen, County of Edinburgh. (Interestingly, Alexander is listed in the marriage record as age 49, a widower.)
1891 census, Alexander, Elizabeth and their two children were living in Uphall, West Lothian.
27 MAY 1891, son Charles Noble McBurney was born to Alexander McBurney and Elizabeth Noble McBurney in District of Buckhorn, County of Fife.
23 SEP 1896, Alexander McBurney (“McBirnie”) was killed in a train steam accident in parish of Kirknewton and East Calder, County of Edinburgh. The death record states that Alexander was married to 1st Sophia McNeilly and 2nd Elizabeth Noble.
24 FEB 1899, Sophia McNeilly McBurney (“McBirnie, widow of Alexander McBirnie, brickburner,”) died in Eastern District of Old Monkland, County of Lanark.
1901 census, sons Alexander and Charles McBurney (“McBurnie”) were living with their uncle, aunt and grandmother (“Elizabeth Noble”) in Beath, Fife. No record was found of their mother, Elizabeth Noble McBurney, or their sister, Elizabeth McBurney.
16 JAN 1916, son Alexander McBurney (“McBurnie”), a Lance Corporal in the Black Watch (Royal Highlanders), died of wounds in WWI.
26 MAR 1918, son Charles Noble McBurney (“McBurnie”) married Grace Kirkwood Phillips in Inveresk & Musselburgh District, County of Midlothian. The marriage record states that both of Charles’ parents were deceased.
18 APR 1918, son Charles Noble McBurney (“McBurnie’), a private in the Black Watch (Royal Highlanders), died in WWI.
After an extensive search of marriage, death, census and immigration records, nothing further has been found regarding mother Elizabeth Noble McBurney or daughter Elizabeth McBurney.