Why not contact your council? Apart from anything, I would imagine the rules differ depending on the local authority.
I think in general the answer is likely to be "no": you'll only get the permit for the road you actually live on (and you need quite a bit of proof for this).
One option you might be able to do is to get two or more streets combined: on our road we now have permits that allow us to park on any of three roads (they're short and all joined together off a main road). This took quite a lot of interaction with the council (Dublin City in our case). Before, there were separate permits for each street, which suited nobody as they'd each fill up at different times. However, in that case all the streets involved already had permit parking, which doesn't seem to be the case in your situation.