Neilston Cemetery, Barrhead. Built in the 1870's since most Church graveyards were running out of space. You may think it odd that it is called Neilston Cemetery since it is actually closer to Barrhead and Gateside. But you have to remember that both of those places were within the Parish of Neilston. The cemetery is still in use today and has been expanded to the next field in recent years. Map location.