Who I have the same questioned this topic

That would depend on what type of an "upgrade" you are going to do.  If you are going the device payment plan route, then you would need to do that AFTER the assumption of liability, as device payments can't be transferred to another account.

Buying a phone on a two year contract (if that option is still available to you, and it may not be the most cost effective way to upgrade anyway) should be done BEFORE just to ensure you don't get extended during the AOL process.

If you are simply buying a new phone outright, it can be done before and after.

What plan are you on now with your parents?  And what plan does your husband have?  That may affect when and how you "upgrade" as well.