The VRT exemption is for transfer of residence, which it sounds like you have already done. The burden of proof for this is that you owned the car and have paid insurance, road tax, bills, income tax, etc. in the UK for at least six months and you are now moving to Ireland - so you need to show a job offer, residential accomodation receipt or similar. The guides will let you know what documents you need to provide.
You would be unwise to drive a UK-registered car in Ireland for any length of time if resident here, this is being increasingly enforced these days. You can forfeit the car if caught, you will as a minimum have to pay the VRT due and a penalty.
Unfortunately using your parents address in the manner implied would probably be seen as tax evasion. Customs and Revenue will have seen all the attempts to evade VRT possible.
You can, however, console yourself that buying a UK car and paying the VRT should work out cheaper than buying the equivalent in Ireland. Income tax is a lot less, too.
Welcome to Ireland!
SSE