It depends on family dynamics. My parents are fine with me letting myself in. I’m fine with them letting themselves in here as well. Now before they moved close to me, I would always tell them when I was coming into town to visit.

That's definitely not the truth in every situation just because it may be for you. In my culture, If I were to even imply that my childhood home is not my home anymore, or act as though I can't enter whenever I want, my mom would be devastated. My mom has worked very hard to make sure I always feel like it's still my house and I am welcome whenever, without asking.

Home in the sense of a place where you feel at home, of course. Being welcome, of course. But just entering without any forewarning just is disrespecting your parent's privacy, no matter how welcome you are there. It's not much effort to call and say you're at home and it prevents all kinds of unwanted situations. It's great that you have this good relationship with them, but I a basic level of privacy should be granted no matter how close you are.