I have never been on a cruise so I can't really say much all I can say is this:
If your parents don't want to go then it is probably best if you just ask them to think about it. Keep pushing them, they will just get annoyed. Give them a list of positive things saying how careful you will be, and you will check in with them every so often, ect. But still tell them you understand why they are worried and that you will take the necessary precaustions. If they still say no, give them time. They are the adults, it is their decision. Keep asking them, it will only show them maybe you are not mature enough to be on the cruise by yourself. ** I am not calling you immature, I don't know you, I am just saying, bugging them is not going to show them that you can handle a cruise***