While you live in India, if what this article about parent-child relationships in India states about parents not wanting to stop treating their kids as kids, even when they're 20 or 30 years old, is true, I think my advice could suit the cultural context, even though I'm Dutch.
I'm a Dutch adult still living with my parents, and while I'm not perfect at living with them, there are a few things you need to realize in order to make this situation work:
I am a college student (at home due to the pandemic), while she is in middle school.
Neither of these really matters. You and your sister are both your parents' kids, living in your parents home. So, the first rule of living in your parents home: Whatever you do, remember it's their house. They basically get to do whatever they want in it, and as such no matter how firm or polite you are, they are free to ignore your requests.
Once you get older, you might end up being treated more like a roommate or houseguest. But still: Their house, they get the final say in whatever happens in it. You can make requests for things to change, but it's still up to your parents to decide whether or not they want to make the necessary changes to accommodate you.
For example, I love sleeping late on weekends. I can ask that everyone is quiet in the early mornings, but it's up to my parents to decide whether they accommodate me. So sometimes parents do some less noisy chores on Saturday morning, on other days they do start early because they have more to do, or they leave the noisy chores (like laundry and vacuuming) for me to do later in the day. It's all a compromise, I'm essentially still their kid in their home.
After the initial request, I am asked to go back to my room and not interrupt their activities (which is weird, because they interrupted me first)
This to me suggests you need to work on your timing. If your parents interrupt you by screaming at your sister, perhaps the best idea isn't to go interrupt them with the message that their screaming is disturbing you.
Talking with your parents, especially about things that are known to be a bit of an issue, is best done at a moment when they're not already worked up and screaming. Pick a quiet moment. Moments like afternoon tea/coffee with my mother or a drink in the evening with my father work very well for me. Basically any moment where your parent is relaxed, not busy doing anything else and not already worked up: This helps me avoid 'disturbing' a parent at an inopportune time, which in turn helps both of us to keep the focus on the issue we're talking about and not the timing of the discussion.
I try to avoid moments like dinner or lunch, where my brother is present too. I don't want him to have his time ruined by having to listen to us argue in case things do escalate.
"Please do not scream at her while I am studying, it distracts me and I lose a lot of valuable time,"
As for your message, it's probably too firm. You are the kid, they are the parents. This means you only get to ask, they get to tell. One thing that guarantees a shouting match over here is me telling my parents they should or should not do something.
For a good conversation about this, you might want to start with questions that let them tell their side of the story, instead of telling them to please stop. You could ask questions that would help you understand what is going on, what happened, and why your parents are screaming in the first place. Once you know that, all you can do is point out that the shouting disturbed your studying, talk about how it annoys you and how you can't focus, and hope they understand that you'd appreciate them shouting less.
I don't tell my parents to please be quieter when they go to bed, or ask them if they could be quieter when they go to bed out of the blue. I start a conversation by asking them if they perhaps ever noticed the sound their clothes hangers make when they hang them on the hooks that are drilled on the opposite side of the wall above my bed, then introduce the idea that it's loud and often wakes me up fully, especially if I'm only just barely asleep. Asking first if my parents are aware things are an issue to me often helps me keep conversations de-escalated, since it offers an opportunity for my parents to say they weren't aware, instead of me sounding like I'm demanding or accusing.
But there is no fool-proof interpersonal skill that could make your parents stop. Assuming your parents don't yell at you or your sister over 'nothing', and that your sister has a role in escalating and keeping the shouting matches going, you can't dictate whenever they can have their shouting matches. You might be able to foster some mutual understanding, which will reduce the frequency of times you have to deal with this, but it's very likely that you'll just have to put up with this until you can leave their home again.
