I think the issue depends on whether you are on a a comment URL (e.g. clicking on the timestamp of a comment), or if you are on the post. If I'm scoped into 20517, the "parent" option is indeed a link to 20429. It will hard load to scope into 20429.
But if I'm on the actual 13396 post and click parent on your comment, I am instead 'loading' 13396#20429. Which of course isn't reloading the page but scrolling to a specific header.
I never personally liked the decision where replying or loading a comment took me to a whole new webpage instead of rending a formbox within 13396 and/or loading 13396#[comment ID], but I imagine there were historical and/or technical reasons for the decision. It generally doesn't bother me as I tend to mostly browse comments and only occasionally respond.
I am instead 'loading' 13396#20429. Which of course isn't reloading
the page but scrolling to a specific header.
No, you're scrolling to that via javascript. Previously this was handled by not interfering with the link to that URL thus preserving history and your location on the page.