I think they normally try keep an airline within a certain group of gates, but it can depend more on how long the plane is going to be staying and what the airline has planned for it while it's there - and while the information about who leaves from where is out there it can take a fair amount of sleuthy browsing to find it

For example ...
Here's a chart of NZAA showing where the gates and parking spots are, and
here is a list of which domestic gates are used by which airlines. I can't find an equivalent list for international, just
this which says
Aircraft Parking International Apron - Stand allocation — pre-arranged through Airport Company Apron Tower. Call APRON to confirm allocation.
Having said which, in all our international travels (Aus, Europe via US, Europe via Japan) the actual planes we've departed Auckland on have been Air NZ ones, and they have always departed from gates or apron parking spaces on the Western side of the finger.
Cheers
Gary
The above post is in the public domain and is guaranteed by the manufacturer to contain no references to anything illegal or discussion of piracy, although this signature may contain traces of nuts.