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My father has a 2002 Holden Barina (parked outside) which is overdue for a cambelt, but has only done 85k's. I was wondering what reliable small vehicle could he get for say 10K, which would last him at least another 5 years, with minimal maintenance, as it's a tossup between keeping the vehicle he knows so well (with more maintenance expected) vs getting used to a "new" one again (he's 94), and the hassle involved finding and buying another vehicle? It's had a few age related problems, but it's been mostly very reliable. BTW I'm in Wellington; he's in Auckland, so I can't really do much except online and advice.
This is a difficult choice and the main decision on this would have to come from the driver of the car.
There are some relatively simple and reliable small cars that sit inside the budget.
2012 Honda Jazz/Fit, Toyota Vitz/Yaris, 2012-2016 Toyota Aqua hybrid, or 2012 Suzuki Swift as examples. The benefit of a newer car would generally be that they are in fact easier to drive, use less fuel and could have more safety features than the outgoing car. Once you have a few, best to take a drive and see if it feels comfortable, then get a vehicle inspection.
We have a preferred dealer network that use our services and would be happy to assist your father find the right vehicle. https://www.aa.co.nz/cars/buying-a-car/car-buying-guide/aa-preferred-dealer-network/