OK, checked my copy of Panorama, and the "Air" attributed to Carolan is the same as the Paret tune: in other words, it's another (simpler) version of As Sylvie Was Walking - and the melody line is pretty much note for note the same as Paret's (though I haven't compared them bar by bar).

Incidentally, Women of Ireland in that book is erroneously noted as being traditional Irish, but in fact it was composed by Seán Ó Riada (Mná na hÉireann).