You're overriding your $query variable in destinationSearch method...

public static function destinationSearch(){

    $query = new ParentRegionList;
    $query = DB::connection($query->connection); // $query isn't ParentRegionList instance anymore...
    $query = DB::table($query->table)->where('RegionName', 'New York')->get(); // $query->table is null? And DB::connection has no effect anymore...

    $query = DB::getQueryLog();
    dd($query);
}

Probably, what you need is something like this:

public static function destinationSearch(){

    $parentRegionList = new ParentRegionList;
    $query = DB::connection($parentRegionList->connection)
                        ->table($parentRegionList->table)
                        ->where('RegionName', 'New York')
                        ->get();

    $log = DB::getQueryLog();
    dd($log);
}