How do I separate my iCloud account from my parents' account?

How did they become mixed? Do you instead mean that you wish to separate your content from the account that you were sharing with your parents, or do you mean you had separate accounts but at some stage the content was melded? The latter typically happens when you have a device that was signed into one account and that device was not properly erased before a second person signed into it.

If you had a shared account, create a new account for yourself. This article explains how to create a new Apple Account --> How to create a new Apple Account - Apple Support Once on the page, select the kind of hardware you are using. If asked if you want to erase the iPhone, say no. Once you are successfully signed into your new account you will later have to go through and delete all the things that you do not want.

If instead your data became merged you will now essentially do the same unwanted data deletion steps. This will mean going through everything on the iPhone. It should have been reset properly before you signed into it but it's too late to do that now. The steps in this article should have been followed before you signed in:

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

To make sure that no other data is shared between you new Apple Account and the old account, you should erase your iPhone, set it up as a new device and start using your credentials after that. This is the only way to make sure, that no data is getting mixed up in the future.

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

You can use Family Sharing to share content and apps bought by other members:

How to set up Family Sharing on your iPhone, iPad, or Mac - Apple Support