http://www.example.com/my_dir/value/

http://www.example.com/my_dir/filename.php?parameter=value

Case #1:
location /my_dir/ {
  if ($args ~* "filename.php/?parameter=value1") {
      rewrite ^ http://www.example.com/my_dir/$arg_param1? last;
  }
}

Case #2:
rewrite ^/my_dir/(.*)$ /filename.php?parameter=$1 last;

Case #3:
location /my_dir {
    try_files $uri $uri/ my_dir/filename.php?$args;
}

Case #4:
location /my_dir/ {
    rewrite ^/my_dir/filename.php?parameter=(-+([a-zA-Z0-9_-]+))?$ /filename.php?parameter=$1 last;
}

try_files my_dir my_dir/ /my_dir/filename.php?$args;