Assuming our original querent is talking about performing for Irish American seniors for St Patrick's Day, rather than Irish seniors living in Ireland for St. Patrick's Day (the list of songs would be dramatically different between the two):

My Wild Irish Rose
Eileen Aroon
Harrigan
Arthur McBride/Bold Robert Emmet (the two are pretty interchangeable)
McNamara's Band
How are Things in Glocca Mora
Did Your Mother Come From Ireland
Erin Go Bragh
Galway Races
Four Leaf Clover
Irish Rover
Sweet Rosie O'Grady
Too-A-Loo-Ra-Loo-Ra/Irish Lullaby

My set list would definitely include Danny Boy (it is a top 5 favorite of the current dying generation of Irish Americans), but since you said that one was out of the question because of the circumstances particular to that nursing home, of course not.

But I would never do The Parting Glass at a nursing home. Too sad. Nostalgia is what they want (which is why Danny Boy is usually fine). Keep it light, give 'em what they want. It's what they are paying you for, after all. Not YOUR taste in music, but your ability to perform the songs that conform to THEIR taste in music.