I don’t know who does what when it comes to parshios tefillin. The machzor I use says that some say them and some don’t and there always seems to be a discussion about which way we follow. As the old joke goes, the minhag is to argue about it.
In terms of the question of the OP, I’m not sure what the question is. If you are saying the birchos krias shma, then why not say shma itself as well? It is part of davening.