Parking is fairly relaxed in New Zealand as there aren't that many cars for the size of the country. You must remember to only park in the direction of the traffic unless it's a one way street.
Paid parking is easily found whether it's on a meter, in a parking lot or in a multi-storey car park. Parking isn't generally expensive except in the big cities.
You must not park on broken yellow lines, on clearways, bus or taxi stops. Enforcement of parking is done by the police and you may get a fine or, if causing an obstruction, you'll be towed.
EU Blue Badges for the disabled aren't recognised but New Zealand does its best for less mobile drivers. Find a convenient spot in an attended car park and ask for assistance; it'll be willingly given.