Upgrading the bed means...
1. You need a new bed - you can use just the frame but then you will not have a headboard.
2. You need a new mattress and box spring. Decent ones will cost about $ 600.00 for the set
3. A new mattress pad. These run about $ 30.00
4. A total of 4 pillows You can buy these for about $ 7.00 ea for inexpensive ones.
5. New sheets
6. New bed skirt
7. New blankets/or Comforter set (a comforter normally set comes with sheets, shams and a bed skirt). These can cost anywhere from $ 40.00 on sale to several hundred. Figure $ 80.00.
So the cost is About $ 700.00 to get you started...Give or take.
A twin is 39" wide and 75" long not counting the headboard and the sideboards.
A Full/Double is 54" wide and the same length. So 15" (15 inches wider)
First off get a ruler and see how much more space it will take up in your room. Make a line on the carpet with chalk (it will vacuum up) see if you will have enough room with a larger bed.
Now $ 700.00 isn't cheap. I don't know if your grandparents or aunts and uncles can be part of this. But it may help if they can.
Another thing to consider is a queen is another 6" wider then a full/double 60" wide and 5" longer 80" long. The queen will grow with you better then a double if you have the room. The cost for all the bedding is about the same. It can be your first bed when you move out in 6 or so years. Mattresses are good for 10 years - longer if it's just one little girl sleeping on it.
So how do you convince your mom and dad. Tell them it's important to you - you understand it's a big expense. Tell them it's all you want for your birthday so maybe the grandparents can chip in. Show her the chalk line on the carpet to explain how it will fit. Offer to help in anyway you can to repay - extra chores or taking a much lesser present next year or even this year if they need to wait a few months to get the money together. Tell her how it will grow with you.
Parents aren't unreasonable as a whole. They do have financial limitations and it is just after the holidays. Maybe if you offer to wait until spring and just get the bedding now everyone will be happy.