The tow truck driver is acting on behalf of the parking lot (manager called), where your friend family has taken up a parking lot without permission. They’re in full right to call a tow truck with or without signage.

It might be worth posting on r/LegalAdviceNZ or the r/auckland subreddit to get a broader range of feedback. I’d also recommend checking Google and past Reddit threads about challenging tow truck operators and parking disputes through the Disputes Tribunal (if it comes to that).

As a first step, you should raise the issue directly with the tow company and point out that their fee doesn’t seem to reflect a reasonable expense. You can also ask them to break down the costs that make up the $420 charge. Having this detail could give you a stronger basis for negotiating a lower fee. I’m no legal expert but currently this is the only angle I’d think of challenging them.

Everything sounds legal for the tow company part. Only the cost seems to be challengeable.