I think you'd show your independence and responsibility if you maintain full time employment for a long period of time (or full time schooling) and preferably a career of sorts. Something stable and something you can support yourself for life.

I don't think doing more chores at home will substitute for true independence. Maybe money that are spent on pets could be spent on stuff like taking classes/certification/training etc Plus having to do so many household chores and care for tons of pets might keep you from getting a career and proper independence

Since it's your parents house it's ultimately up to them to have pets or no. I am also concerned that pets that are already there, aren't allowed in the house. That's kind of crazy. There are too many pets as it is if there are problems with them peeing in the house or not getting along with each other. I'd be afraid to have more pets and them being kicked out too.