Are my goldfish going to be okay?

[Nemesis]

Alright here's the story, I've recently bought 2 goldfish and placed them into a mediocre sized bowl with some rocks, and a live plant on a tray. The goldfish are gulping on the surface, yes I know it's trying to get oxygen but the thing is when I explain this to my mom, she says that it's fine and I'm worrying too much and it's natural for them to do this and refuses to buy an aquarium. So what can I do for my goldfish? Can an air pump/airstone fit in my mediocre sized fishbowl? My fishbowl are about 3-4 times bigger than your normal fishbowl. Also, how can I persuade my mom to buy an aquarium for my goldfish?
Yes, I am using something that kills chemicals in the water

 

[Sunday P]

Goldfish cannot live in any bowl. They are huge when adults like 10 inches or more. They need at least 10 gallons per fish, airator, filter and in my opinion heater too. They are gasping due to low oxygen and ammonia build up.
For now change water everyday, and yes you should add an airator ASAP. A small sponge filter will also help, and if the fish are small (under 2 inches) then they can make it this way for a little while. Also put in a real plant will help also. If you intend to keep them they need a proper home like a 20 gallon tank. They will need this fairly quickly. Try doing chores to raise the money yourself and you take care of the tank. It's a good life skill. Do some research. Any reputable web site will tell you a bowl is no place for a goldfish. A betta would love a nice big bowl like that, goldfish however do not. Goldfish in bowls live weeks, maybe a month or two if they are super tough. Goldfish are supposed to live 10 or more years if kept well. The oldest was in its 40's! Putting goldfish in a bowl leads to a short miserable life for the poor things. They die of their own waste products, how sad.