[David L.]

Remember that what we say is meant to convey information to another person. You have three pieces of information in your words: the country Peru; an action or state of 'living', and your 'parents'.

Are you going to tell me about Peru? If so, then that's the most important thing to let me know, 'this is what I am speaking about', and so it would go first.
But it isn't. You are talking about your parents. So parents should go first. This is the noun, and the verb 'live', which tells us about some action or state of your parents, comes next.. We then add other words that modify the verb or add more information.
So: my parents +living + and they live in a specific place, in a country called Peru.
"My parents live in Peru."
compare:
"America seems such a wealthy country. In Peru, where my parents live, most people are poor."
compare
"To live in Peru these days, with inflation at such record heights, is not easy. It certainly keeps my parents poor."