It is possible to share templates between a parent account and sub-accounts, similar to sharing contacts. Shared templates are read-only to the accounts to which they are shared, and cannot be edited except by the account which created them.
To share a Template with one or more sub-accounts, go to Messaging, then Templates, and click on the three dots next to the template you wish to share, where you can then select Manage sharing:
Select the sub-account(s) with which you want to share the template and click Share:
In the sub-account, shared templates are denoted by the "shared" icon:
Note that shared templates do not have the three dots next to them, as they cannot be edited by the sub-account.