I would get it apprised, if you have not done so already. Remembering that just because it is apprised at a certain price does not mean you will get that amount when you sell it.
When you bought it from grandma was she going to sell it anyway to family or unknown person? To me if she was going to sell it to whomever could pay then you are doing nothing different then her. The only thing if you are looking for MORE money then what you paid for the guitar it can come across that you were out for making money on something that has important family meaning. That can tear any family apart, so be careful how you go about it. Honestly if you are saying you can afford school AND maybe a house down payment it kind of seems you are selling a potential family heirloom just for the money.
How much did you pay for it (hundreds, thousands)? Offer to sell it for the same price you bought it at to any family interested. Right now it seems you want to make quick money, even if it is going to a good cause. If someone did that in my family with a potential family heirloom I too would be a little upset.
I do agree that it is yours to do with what you want, you bought it, it is rightly yours BUT if you do sell you NEED to pay back the $200 that the parents fronted. Also you said you sold your car that your parents gave you PLUS they fronted the $200 really your PARENTS bought the guitar since you used their gift to buy it... the more I think about it the more it is about getting more money.
Regarding school you can work more hours, apply for grants, take out a school loan. I got hardly any finical aid at a mid price private college and still made it work without selling valuables. To me if this guitar was so important for you to sell your car & ask parents to front you money then you may regret it later if you do sell.