How would I get WordPress to parse /mypage/area/value as /mypage/?area=value?

Question

add_rewrite_rule is very handy for this use, first add your custom rule for this:

add_action('init', function(){
    return add_rewrite_rule(
        '([^/]+)/area/([^/]+)/?$',
        'index.php?pagename=$matches[1]&area=$matches[2]',
        'top'
    );
});

and you already registered the custom query variable, next is (because you’re in development) go to admin > settings > permalinks and save settings to flush the rewrite rules.

Now to get the custom area, just call get_query_arg('area') which will return the value when WP_Query is ready.