I see no reason to tell your parents. I havent told my parents and they are 1.5 miles away, not 6000. If they do ask, be honest. If they don’t let them assume what they want to, noone gets hurt that way.
My parents “know” that i’m otd but my dad doesn’t push it. we have had a great don’t ask dont tell policy for > a decade. My mother asks too many questions (of which i am honest). when she doesnt like the answer, or more often says “don’t tell your father it would kill him” i usually respond something like “if you don’t want to know, stop asking” or “It WOULD kill him, thats why he has the sense not to ask”. In the meantime, my father is amazing. he will repeat a good vurt if he hears one and we have many intellectually stimulating conversations about science and history. Our relationship could not be better. and before someone argues that the relationship is based on a falsehood, i will preempt to disagree. Life is not all about religion and religious observance. All the things you all have mentioned about davening 3x a day, or saying tehillim or doing tshuva has NOTHING to do with your relationship with another human being.