Somewhat surprisingly but sadly, the answer is no.
On first reading I thought "Elite Partner" referred to elite members of partner airlines, but it's actually a weird Air NZ status. Star Alliance elite members only get the usual allowance of one guest, and only children under 2 are free. You can, however, bring in paying guests at A$30/person (in Sydney, prices may vary elsewhere).
Source: Staff at the Air NZ lounge in Sydney, confirmed by a visit today with kids in tow.
For what it's worth, the neighboring Singapore Airlines lounge (also Star Gold, but much smaller) does not allow paid guests, period.
Update: The last four times I've tried my luck at an Air NZ lounge by stating "I have a guest and two children", the entire family was let in for free twice (both times at SYD), and twice I was asked to pay for the kids (SYD and WLG). Your mileage may vary, and my kids are way above 2 years old at this point.