This is why I'm glad I don't have kids, because I would have even more animals than I do currently! LOL! I was able to get a dog, then a couple hamsters in my mom's house, but I had to write a page long essay about the complete care of the animal and how owning that animal will help me in my career in the future. But I ultimately had to wait until I moved out of my mother's house before I was able to get what I really wanted and now i have a dog, 3 cats, 2 rabbits, snake, 3 tarantulas, 2 rats and fish tanks, and I might be getting 2 more snakes soon
Just wait it will come. I'm not suggesting going hog wild like I did either, LOL! As it's getting to be a lot of care, but i love it!
Caring for animals has to be something you truly love doing and if you're not helping with the animals that are currently in the home (what i gathered from when you said "they're not even mine"). I probably wouldn't let you get a rabbit either until you showed you can handle and were interested in the animals that were already there. Even if it's something easy as refilling the guinea pig hay, water bottles, helping with the weekly cage cleaning, etc.
The essays my mom made me do, made me be a research fiend! Now I can tell you a ton about care/husbandry/feeding/etc about almost any animal I have owned because I constantly research every aspect of their care, especially diet.
Make sure you respect your parent's wishes.