Sara,
First off, is there an extra room available or would the family have to move to make one available for you? If there is no extra room, it is doubtful that your parents would go to the expense of moving for this reason.
If another bedroom is available, it makes sense for you to have your own room. Perhaps your parents dread the idea and effort of moving beds and other furniture after working all week.
Maybe another bedroom is not sitting unused, but an attic, basement, dining room, office, enclosed porch, or other area could be transformed into a bedroom space. It would not have to be forever, just until one of you leaves home for a university or is out on their own.
Consider enlisting your sister into supporting your idea. Would she like a private space of her own? Do you have separate interests? One practices a musical instrument, the other does not? One really likes privacy to read or study, the other likes to visit with friends? If both of you are asking for this and are reasonable and cooperative, maybe your parents would go for it?
Before pushing too much for something, always consider the viewpoint of the person or people with the power to say yes or no.
Do they think it is too much work?
Do they worry about the expense of buying additional furniture, paint, etc?
Do they think the idea of privacy for children is silly?
Do they think you will get your own room and keep it looking like a pigsty, therefore one more room to fuss at kids to clean up?
Does your room currently resemble a place to raise animals? Is the thought of cleaning it just overwhelming to them?
Think about what their objection will be and plan solutions for any potential problem to counter those. If your room is currently messy, clean it thoroughly. Draw a plan on paper showing where your things could be in the new space. And where things that were there could go or be stored.
If it is a space and money issue, you have to accept that. Then go to Plan B. How can you gain more space in the room you have? Is it large enough for a divider like a hanging drape or bookcase? Plants? Agreement with your sister about each giving the other at least an hour or hour and a half of alone time in the room during the evening? Form an alliance with her to get what each of you need.
I shared a room with my younger sister until I moved out and yes, there were times I would have really enjoyed space to myself. But it was what it was and I just got out of the house more, visited friends, walked around, sat in another room or outside. You can survive anything but I do hope you can get a space for yourself.