Although it’s possible to do it the way you’re wanting, you might just find that it’s more of a hassle than it’s worth. You can read more about changing your sites URL in the WordPress codex.
Although, as hazephase has pointed out, you can change the URL in Settings, you’ll probably have a lot of cleaning up to do when you make the move. Consider for example you add an image to a page, WordPress uses the absolute path of the image, i.e. https://www.yoursite.com/wp-content/uploads/… which in your case will be https://www.themeparkconnect.com/wp/wp-content/uploads/…. These sorts of things won’t be fixed by changing the site URL in the settings. Which means you’re either have to fix them page by page or dive into the database and fix them there. All that said, you could also use relative paths when designing the site. Your HTML already uses relative paths.
I’d recommend installing WordPress directly into the public_HTML folder and use the “coming soon” plugin I suggested earlier. It’ll be really easy to copy the HTML into the plugin and have it manage the coming soon aspect of your site.
Another thing to consider is that if search engines index your site before you move everything out of the WP folder, they are going to have a lot of broken links to deal with the next time they scan your site.
If you decide to go with my suggestion, make a backup of the contents of public_HTML before you install WordPress. If things don’t work out, you can always delete everything and restore the backup.