Qantas are squarely in the wrong, for a couple of reasons.
1. As a supplier, if you sub-contract the job to someone else, the buck still stops with you. Unless the agreement was the the pax was going to pay for the chauffeur themselves, Qantas is responsible. Since the chauffeur was being paid by QF, and it's part of the ticket they had purchased QF don't get to pass the buck on this one.
2. If a service fails to meet expectation, the supplier is potentially liable for loss or damages. The costs incurred by the OP are not unreasonable and therefore the OP would be entitled to the full amount of the parking, plus a per KM allowance for fuel and reasonable wear and tear on their own car. Long story short, what is reasonable is very situation dependent, but what the OP did does not seem unreasonable to me.
3. Taken straight from the ACCC website -> "Suppliers must not mislead you about your rights to compensation by stating they are not responsible for any losses you might suffer during or after using the product or service." So QF stating they have offered something, take it or leave it doesn't cut it.
and
4. Compensation for damages and loss
https://www.accc.gov.au/.../compensation-for-damages-loss
All this other talk about child seats and legal issues around such seats in taxi's is irrelevant, as is the talk about what risks you might be willing to take with your own children. The other thing that is irrelevant is the fact that the OP waited 30 minutes before speaking with QF. The fact is that the service was not provided at the agreed time and thus set off a chain of events.
Finally Qantas can have all the T&C's in the world, they can not override consumer law. Reading over their T&C's I'm a little surprised they don't have a severability clause, clause 17.b kind of deals with the fact that section 17 may not be entirely legal. I'm also a little surprised that they don't have a jurisdiction clause, as that can make it kinda messy for QF if something goes wrong.
PS, not a lawyer, but I am someone who personally takes companies to task over consumer law.
So following on from the above, if I were the OP, I would write one final letter explaining that you are not happy with the compensation offered from Qantas, here is the justification as to why, and that they have 14 days in which to comply or you will be escalating the issue to NSW Fair Trading. Also put some links into relevant sections of the ACCC website to emphasize your point. Don't use emotion, just provide a timeline and the facts.
That 14 day cut off is important (provide an exact date), as it gives them a firm deadline in which to respond.
Good luck