I took a glance at the online Irish phone book (which only covers land lines) and Tarpey is much more common than Tarpy. The surname is mostly found in Galway and Dublin but there are Tarpeys in Roscommon, Mayo and Sligo.
Can you identify your Tarpey ancestor that left Ireland and when he left? Also his age? Maybe we can find out more info on your family in Roscommon.
The other question is whether you have taken a DNA test. I don't recommend the Y- test which tracks your pure paternal Tarpey line but rather a more general test that is available thru Ancestry and other companies. You may get some close matches on your Tarpey line