In general, No. You are a separate legal entity from your parents. If an account is in your name, it belongs to you and not your parents. I think this applies even if you are a minor.
The only way the bank would be normally be allowed to do such a thing would be if you agreed to allow them to do so, such as if you already agreed to provide a guarantee to your parents, or if the account was held jointly with your parents.
Even if you personally had two accounts, one in the name of a business that you controlled (a separate legal entity) and your personal account, they wouldn't be able to take money from your personal account. Usually in these situations however, the bank often requires a personal guarantee to open the business account.
One exception to the rule is in the case of impropriety. If the bank was asking to be paid back, and then suddenly you were 'given' the entire balance of your parents account. It could be argued that this action was just performed to prevent the bank from getting their money back (a kind of fraud). As a result the bank might be able to 'trace' the money. It doesn't matter which bank the money is held in for them to trace it, but if is in a different account at the same bank, it requires less legal process for them to get hold of the money.
These are general rules for "common law" countries (USA, UK, Canada, Australia, NZ, India etc.). Your specific situation will depend on the local laws in your country and any agreement you signed with the bank when you opened the account. A local lawyer would be able to advise you more precisely.