I understand your pain... Believe me...
Try telling your mum that it'll save money and time in the long run to buy a bigger horse. You are most likely going to outgrow a 14.2hh pony, if you are as tall as you say you are. I am very tall and even I couldn't wrap my legs around a 13hh pony and touch my toes together under the belly! You are definitely tall enough for a bigger horse.
If you DO outgrow a 14.2hh pony, you will have to sell it and find another taller horse. This is money, time and effort down the drain. Try putting it to her like that. Also I have gathered that you are mostly leg. Not unlike me. If you have long legs, you take up more of the horse, have more control of the horse, and its generally easier to control a taller horse as you can wrap your legs around it easier than someone with shorter legs. You won't look small then on a 16hh horse. Far from it. Unless you buy a really wide draught, which I'm gathering you won't as you want to do show jumping, and show jumpers tend to be more blood horses with a narrower frame.
If she still doesn't budge, try a compromise. The horse doesn't have to be as tall as 16hh... Maybe look at some 15.1hh, and 15.2hh. She may be happier with that. Some people, I find, think of 15.3hh as being a nice height, then they hear 16hh and go Ahhhhhh! Because in their mind it sounds huge! I was the same and maybe that's what you're mother is like.