Parents deposit the money in the student's name and have total control over it. The money deposited gains interest at the current interest rate, which currently is about 2.1 percent. Savings accounts are usually for long term savings. Parents usually do not put too much money in saving accounts for their children because most of these accounts transfer all the rights to the money to the student once the students becomes 18. This means students can spend the money in whatever way they please when they turn 18, which sometimes does not matches the parent's wishes.
Parents have to be in charge of the account because of legal issues.
The risk factor is low for saving accounts. The money is protected by the federal reserve, which means that even if the bank loses all its money, you will still have your money. This program was implemented after the Great Depression because many lost their money in the banks after the stock market crash. Since the risk factor is so low, the return rate is low too. Although you are gaining money every year, you could be losing purchasing power with your money.