The Fiddler's companion says this:

AKA and see "The Parting Glass." Irish (?), Air (4/4 time). A Minor. Standard tuning. AABB. The title refers to the Scots poet Robert Burns, who wrote his song "The Parting Glass" to the tune.

The tune was first notated in Scotland in 1625. No "Irish(?)" about it. It was for the mandour, which is tuned ADADA, and it was in D mixolydian.

Burns's poem to the tune is not called "The Parting Glass", nor do those words occur in it. The version of the tune Burns knew was in E minor as published in the Scots Musical Museum, which he co-edited. The "Parting Glass" title was popularized, and possibly invented, by an Irish pub-folk-rock group of the 1960s.

The first known text for it was anonymous, published by David Herd in 1776.

More about it near the end of this, part of my collection of the music of Edinburgh.

The Fiddler's Companion is the National Enquirer of musicology.