If I sign out of my Apple ID on my iPhone to sign into my old Apple ID, will all my iMessages be deleted?

If you were using your parents ID, unless you had your phone number on their ID, the messages probably aren't tied to the phone number.

If 13 years old or older.

Apple ID - create a new.      

Apple ID – Create         

If under 13 years old, you will need to have your parents set up Family Sharing to create an ID.

Family Sharing and Apple ID for your child.      

Family Sharing.         

Family Sharing - Set up.        

When changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. Also set up Family Sharing between the IDs to keep access to your apps/purchases. 

Set up - your child’s iPhone, iPad, or iPod touch 

How to switch Apple IDs on your iPhone or iPad.      

How to transfer all files from one iCloud account to another.        

What to do after you change your Apple ID or password - Apple Support.     

You won't have access to them. You could sign out of everywhere but Messages if you still wanted to be able to access the current Messages.