Why do you need to destruct whole your HTML?
Why do you use DOMParser?
Yes, of course, when you remove element from DOM you also removing listeners and when you adding new elements into DOM they don't have any listeners, however, you can MOVE elements in DOM and save attached listeners, but why do you need to attach event listeners on something you removed? You NEVER remove element from DOM if you need it. Just init() widgets you inserted into DOM later or use DOMObserver API to do so automatically.
So, answer to your main question is obvious - you have to attach event listeners exactly the same way you attached them first time.
However, I very appreciate that you are using vanilla JS and you may look into BunnyJS and join Gitter chat room where I would be able to provide your with some tips on how you may write simple and on the same time powerful, maintainable, modern code using nothing but plain JavaScript.