would it make sens to run this in parallel?

makes sense
but i have no idea how to implement this))

How this could be runned in parallel?

Avoid consuming the response: When you access a property (calling a getter) the response blocks until the response is fully processed.

once again, I don't know if it's better to run theses requests in sequence or parallel.
(Maybe run too many requests in parallel have worse performance) => you need to check AWS recommendations for this.

to process requests in parallel you should do something like

for (...) {
  $responses[] = $client->uploadPartCopy(...)
}

$success = true;
foreach ($responses as $response) {
  try {
    $copyPartResult = $response->getCopyPartResult();
    $parts[] = new CompletedPart(['ETag' => $copyPartResult->getEtag(), 'PartNumber' => $partNumber]);
  } catch (\Throwable $e) {
    $success = false;
    break'
  }
}

if (!$success) {
  $this->abortMultipartUpload(['Bucket' => $bucket, 'Key' => $key, 'UploadId' => $uploadId]);
  foreach ($responses as $response) {
    try {
      $response->cancel();
    } catch (\Throwable $e) {
      // ...
    }
  }

   throw ...;
}

Got it!
Seems like the only limit is 10 000 connections (equal to parts)
https://docs.aws.amazon.com/AmazonS3/latest/userguide/qfacts.html
Anyway, think that 10k connections, even without body, it is too much.

AWS SDK has concurrency option for such copy
Will see how to implement it here