Murray Shoolbraid has identified the "Peri-Periwinkle" song and the one above as parodies of a song by Robert Tannahill titled "Jessie, the Flower o' Dunblane" (see here for full text):

The sun has gane down o'er the lofty Ben Lomond
And left the red clouds to reside o'er the scene
While lanely I stray in the calm simmer gloamin'
To muse on sweet Jessie, the flow'r o' Dunblane.
How sweet is the brier wi' its saft faulding blossom
And sweet is the birk wi' its mantle o' green
But sweeter and fairer and dear to this bosom
Is charming young Jessie, the flow'r o' Dunblane.
Is charming young Jessie, is charming young Jessie
Is charming young Jessie, the flow'r o' Dunblane.

"The Flower o' Dunblane" was first published in the Scots Magazine, March 1808.  Matthew Tannahill, the poet's brother, said it was composed to supplant the old (and suggestive) "Bob o' Dumblane".