You can get a paper round at 11. But you can't work at somewhere like a supermarket til you're 15.
I think that website it saying the neighbour next door can hire you to mow your lawn at any age, but you can't work for an actual lawn mowing company until you're 15. (Just a random example.)
I got a job at a supermarket when I was 16 and consider myself very lucky as I know a lot of people who don't have jobs and clearly need one (only other option for them is going on the dole.)
Friends' jobs include Bunnings, play centre, doughnut shop, supermarkets, clothes shops in the mall, coaching a particular sport they are good at.
It helps being in a city like Auckland or Wellington though. When I was 13 I don't think anyone was bothered about having a job. That seemed to change as I got to 16+.
How old is your older son?