You sound like me not all that many years back. If you don't mind my asking, are you old enough to work? My parents got a bit more flexible once I was old enough to foot the bill for my hobbies. When I was able to pay for it, they let me have a reptile aquarium for a bit, and they never really took issue with fish so long as they weren't constantly dying or making too much fuss.
Another outlet if you're old enough to work could be working in a pet store or feed store if you have one around, depending on whether your interests move more towards exotic animals or farm animals and what is local. That way you could learn from others, read in their book sections and get a nice discount on research materials and eventually pet related supplies if you're lucky.
Over time, by showing that you are responsible and working to take care of your own needs, then it might show them that you are also capable of meeting an animals needs.
Pet sitting and dog walking might be a way to get your fur fix as well even if you aren't old enough to get an "official" job. Mucking stalls for a few bucks or in exchange for instruction is another good way to start learning about horses and animal health and nutrition.
Edited by HnkyDnkyZZFarm - 6/4/16 at 5:43pm