URL parameters causing 404 on home page, but nowhere else

I was having this same issue. I found that the issue was caused because I had set the front page to display a static page for the front page and another for the posts page. (Set in Reading Settings)

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However, my posts page had been deleted. In this case because we decided not to have a posts page any more. The URL variables were trying to switch the home page view to the posts page that did not exist.

I fixed this by updating my template to use front-page.php (rather than a page template) and changed the setting so that the front page displays Your latest posts.

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This solution worked for me because I did not need the posts page.