The area below the lawn where you wish to excavate would be common property, and as you’re in NSW (or your IP address is) and your proposal would comprise a change to that common property by the construction of a new structure below it, you’ll need to obtain the prior written consent of your Owners Corporation (O/C) by submitting a Motion supported by detailed plans, trades license, and liability insurance details to a General Meeting, seeking consent to your proposal via a vote in favour by a minimum 75% of those present both in person and by proxy (i.e. a special resolution) calculated on the basis of their lot units of entitlement (UOE).
Your Local Council would require the O/C to make a notation on your DA or BA to the effect that its consent to the above has been granted.
In the meantime have a chat with your Strata Manager, but more importantly your fellow owners and particularly those on your Executive Committee re the above, and if indications are that you can jump the above hurdle, then you’ll need to concurrently work on seeking that same minimum degree of consent (i.e. ≥75%) for an Exclusive Use By-Law and its subsequent Registration, that as the name implies would need to be legally drafted at your expense to grant you and subsequent Owners of your Lot the exclusive use of the area of common property occupied by your proposed garage extension, and document that you as opposed to the O/C will be responsible for all on-going maintenance and repairs for that area.
Then finally, as the additional parking space would add “value” to your Lot, you’ll need to have that Exclusive Use By-Law set a mechanism by which you make a payment, perhaps annually indexed, that’s equivalent to the levy contribution that you would otherwise pay if your Lot originally had a tandem / 2 vehicle carspace, or perhaps an amount representing the difference between your current contributions and those that other owners with that set-up currently pay.
As for the costs, they would involve those for a Strata Lawyer to draft the Exclusive Use By-Law (EXBL), the costs to convene a General Meeting of the O/C unless of course you submit your Motion to an already scheduled AGM, and the costs to have the EXBL Registered on your Plan’s Strata Title; I’d estimate $2-3K all-up if it goes to an AGM but have a talk with one of FlatChat’s sponsors HERE.
It all sounds complex, but not so for a suitably experienced Strata Lawyer with whom you should also discuss any requirement for a Strata Sub-Division and the resultant costs to, in addition to all the above, add the physical details of your proposed works to the existing Strata Title Plan.