How do I stop sharing bookmarks and browsing history between my MacBook and my dad's Mac mini?
Bookmarks etc are synced via iCloud. To ensure that your two devices aren't syncing in this way you will need to make sure you're not both signed into the same iCloud account. 

To do this:


Go to Apple > System Preferences
Click the iCloud preference pane
You should now see the account you're currently logged into. 
Assuming you're logged into your dad's account, click on the Sign out button at bottom left


NOTES: 


Depending on the arrangement between you and your dad, you could continue using the same account and opt instead to untick the Safari checkbox as this is the setting that is sharing your bookmarks, history, etc.
If at Step 3 you see that you're actually logged into your account, then you will need to access your dad's Mac mini to follow these steps to sign him out of your account.
Note that if you also have an iPhone, iPad, etc you may also want to check your iCloud configuration on those devices as well to make sure you're not using your dad's account etc.


You may also want to read Apple's iCloud page to familiarise yourself with all of its functions etc.