Azure Storage Blob types (CloudBlobContainer, CloudBlobClient, etc.) and thread safety

I am developing an azure application which needs at some point to upload(download) a large amount of small blobs to a single container (more than 1k blobs, less than 1 Mb each). In order to speed up this process I'd like to use multiple threads for uploading(downloading) blobs.

This is routine I use for uploading single blob:

CloudStorageAccount storageAccount = CloudStorageAccount.Parse(ConnectionString);
CloudBlobClient blobClient = storageAccount.CreateCloudBlobClient();
CloudBlobContainer blobContainer = 
    blobClient.GetContainerReference(ContainerName);
blobContainer.CreateIfNotExist();

CloudBlob blob = blobContainer.GetBlobReference(Id);
blob.UploadByteArray(Data);

For each type used in the code above MSDN says following:

Any public static (Shared in Visual Basic) members of this type are thread safe. Any instance members are not guaranteed to be thread safe.

Does it mean that I need to execute following code in every thread? Or maybe I can execute it only once and share single instance of CloudBlobContainer among different threads?

CloudStorageAccount storageAccount = CloudStorageAccount.Parse(ConnectionString);
CloudBlobClient blobClient = storageAccount.CreateCloudBlobClient();
CloudBlobContainer blobContainer = 
    blobClient.GetContainerReference(ContainerName);

I would be really happy to use single instance of CloudBlobContainer in different threads otherwise it seriously slows down the whole uploading(downloading) process.