rsync throwing mkdir: cannot create directory ‘/data/dir_1/dir_2/dir_3/’: No such file or directory

rsync command doesn't create directory tree, so you can do this by perform:

 mkdir -p /data/dir_1/dir_2/dir_3

before rsync command or use --rsync-path options:

--rsync-path=PROGRAM Use this to specify what program is to be run on the remote machine to start-up rsync.

Your command should be:

 rsync -avz source_diretory/ --rsync-path="mkdir -p /data/dir_1/dir_2/dir_3 && rsync" /data/dir_1/dir_2/dir_3

Here is a useful link.