If I am purchasing a park ticket for someone else, can I do that under my account and then assign it to them? Or does it need to be done under their account?
If you will be doing all their planning and they won't have their own Disney account, you can assign the ticket to them under your account.
But if they will want to have their own Disney account and do any of their own planning, then their ticket needs to get associated with their own account, and you can connect with them on MDE to make plans with them.