If I were you I would start by getting a copy of the death cert of your gran's grandfather (John Ivory). You never know what clue this might give. The informant is often the eldest son. John Ivory died in 1926 so, as it is post 1864, you can request the cert from the GRO in Dublin.
To find your gran's parents you need to get two certs.
You need to get your gran's death cert, also from the GRO. Your gran's death cert will show you her age at death. You can then work out her date of birth and (providing she was born in the same place she died) you can get her birth cert from the GRO. Your gran's birth cert will show who her parents were. You can keep going back like this until 1864.
If you have a name, place of death and approximate year you can get a death cert. The GRO will do a 5 year search around the death year you specify and provide you with the cert for € 6. If they can't find the cert, they will charge you €2.
You cannot currently order certificates on line from GRO but you can send a completed form, with credit card details (as you live abroad), by post. Here is the web link to download the form...http://www.groireland.ie/apply_for_a_cert.htm
I hope this is useful and good luck