How to create URL parameters to run custom queries?

What you want to do is access the query_vars and use them.

Base code:

Page to access the fields

$custom_field = get_query_var('custom_field');
var_dump($custom_field);

And the function or custom plugin to save the magic:

add_filter( 'query_vars', 'my_custom_query_vars' );
function my_custom_query_vars( $vars ) {
    $vars[] = 'custom_field';
    $vars[] = 'custom_awesome_field';

    return $vars;
}

yourawesomesite.awesome/?custom_field=white

string 'white' (length=5)