Technically, I don't think your parents have to take responsibility. They've basically allowed you and your family (including the cats) to come live with them and have set certain conditions. While it sounds like those conditions are nearly impossible to meet, they are what they are. Your mother doesn't want to kitty proof her house and your dad doesn't want the litter box on carpet.
Is there a bathroom in the house that only you and your husband use? Can you relocate the litter box there and leave the cats there during the day? I know it's not an ideal environment for them, but it seems it's either that, or try to convince your dad that a plastic floor covering in your room will alleviate his concerns about litter on the carpet (I hate that too, it is hard to vacuum it all out if the pile on the carpet is looser).
I think if that suggestion doesn't work, your only other choice would be find them a temporary home until you can afford to move out.
In answer to your question, it's THEIR house, and those are YOUR cats. The responsibility is yours and your husband's absolutely. In a perfect world, your parents would be totally up for adapting their house to fit your young cats, but that is obviously not the case. Even if they were up to making the changes you'd like, it's still not their responsibility to adjust to accommodate your pets' behavior. Given that, if it were me, the choice would be to let you know to either find another home for your cats temporarily or find another home for you, DH and your cats that will work with your needs.
I'm a bit curious about how you worded your first post when describing your parents:
My mom is one of those hoarders disguised as an enthusiastic decorator. Every flat service in the house is covered in breakable knickknacks and frames.
My dad is like Mr.Clean's OCD brother, so a litterbox in a carpeted room is something he will not allow, ever.
To me, these descriptions definitely slant quite a bit to the negative side. This is your parents' house. I don't actually get the impression that your mother's enthusiastic decorating exhibits hoarder tendencies, nor do I think your father not wanting a litter box in a carpeted room makes him like Mr. Clean's OCD brother (mind you, I am going off the examples you are giving, specifically). Basically, I don't think their wishes are at all unreasonable nor are the practices in these examples unreasonable -- it is THEIR house after all; they can keep it as they see fit so if your dad doesn't want litter anywhere on the carpet and your mom's practice to cover all her flat surfaces with knicknacks works for them, then there are no problems there (if you'd said something more like "we all have to eat with our plates balanced on our laps and holding the drinking glass in one hand because the dining table is covered with stuff" or "Dad makes us take our shoes off outside, rinse our feet off with a hose, then stand on a towel and dry them off before coming inside the house" then I'd get more of where you are coming from with the extreme negative descriptions of them). I will agree, though, that it makes things much more difficult for you, but, really, that's up to you and your husband to figure out how to work around their wishes if you want you two and your pets to continue availing yourselves of their hospitality.
I apologize if I sound harsh. I'm slightly finding myself in the same boat (as far as trying to rein in cats). DH and I bought a house last year, and my two senior cats are very messy creatures (one because he's more like a clumsy dog and the other because he will deliberately make a mess when he is upset with me about something). I love my cats dearly, but I don't want them to take over our house. The decision I've made (that meets with DH's approval) is that I'll get a large pen for them to stay in when we are not home (people do this with their dogs all the time). So, anyway, I'm suggesting that, perhaps, you get a pen for your cats for when you are not home? When you ARE home, well, you really will have to keep an eye on them the entire time; that is the cost of living at your parents' house with your cats.