I wonder if this a Tax fiddle?
I used to employ 3rd parties who were not VAT registered in the UK to undertake work for my company (which was VAT registered).
The invoice from my company to the customer always specified the amount of VAT payable as required by law and that was irrespective of whether any sub-contractor was involved or their VAT status.
The relationship between the principal invoicing company to the customer and the sub-contractor to the principal invoicing company is irrelevant to the customer and is no concern to them.
Clearly, if the sub-contractor is non VAT registered, the principal invoicing company to the customer will in no way be able in law to claim any input VAT from that transaction but because the principal invoicing company is then they are duty bound to charge VAT on the whole invoice which they must account in full to HMRC.
So, the simple answer to your question is a definite no. You are not entitled in any way shape or form to make any deductions from the principal invoicing company's invoice on account simply of the sub-contractors VAT status.
In the same way, the sub-contractor cannot invoice the principal invoicing company for other than his net cost. If he has himself paid VAT on goods he has supplied to meet his obligations under the contract with the principal invoicing company, the principal invoicing company cannot themselves account for the VAT paid by someone else.
It would perhaps be usual practice in such circumstances for the non-VAT registered sub-contractor to identify what supplies he needs and request the invoicing company to acquire them on his behalf in which event, the sub-contractor will only invoice his labour, not parts.
Conversely, if the sub-contractor is himself VAT registered, he will only invoice parts (including any profit element if applicable) at the cost net of VAT and claim the VAT back himself as input tax on his return at the relevant moment (and of course for his labour element). The principal invoicing company will claim such VAT back from HMRC through their VAT return at the relevant tax point.
Seemingly complex, but quite straightforward if you think about it.