DeBarre,
The Hamiltons did not at any time, own the burial ground, it always belonged to the ecclesiastical parish of Dalziel. Anyone, no matter what faith, was allowed to be buried there, the only criteria was that the person had to have lived in the parish.
Parkhead was an area just south of Windmillhill, where Glencairn Street runs now.
In 1808 John Barrie, Windmillhill paid to Dalzell estate £4 for a half-year's rent to Mertinmas 1808. He also paid one year's feu duty of £4.16.9d. (Martinmas, one of the Scots quarterdays)In 1810 Barry's rent and feu duty together was £8.14.8d.
In 1812 John Barry, Windmillhill paid a half-year's rent to Mertinmas and feu duty, total £4.14.9d.
In 1813 John Barry paid a half-year's rent to Whitson 1812 of £4. He also paid £1 for grass rent.
In 1808 John Barry paid property tax of 17/5d. (about 87p)
In 1809 John Barry paid property tax of 18/5.
In 1810 John Barry, Sheills, paid property tax of £8.
In 1784 Thomas Barry was a tenant-at-will at Dalziel for tax purposes. (On the estate of Dalzell).
As for Burngrains, it's where I live!
The modern spelling is Burngrange and it's in the Flemington area, just off Shields Road. (Shiells was Shields Farm).
For a very short period, there was a parish bank in the parish of Dalziel. The records are with the National Archives in Edinburgh. They only cover a few years but it does prove that certain family members, and indeed certain families, were alive and living in Dalziel at the time.
All the following have the surname Barrie and place of abode in all cases is "Windmillhill". Dates are date of 1st and last deposit.
AGNES - 3rd June 1816 - 4th November 1816.
HELEN - 8th January 1816 - 22nd December 1819.
JANET - 4th December 1815 - 16th February 1826.
JOHN junior - 4th December 1815 - 22nd December 1819.
THOMAS - 7th January 1818 - 8th August 1819.
Also -
KING ISOBEL, Windmillhill - 7th January 1818 - 3rd May 1819.
To answer your last question, there was no other burial ground in this parish.