The agent of my partnership is not registered for GST, but does have an ABN. Can this agent still issue tax invoices on behalf of the partnership with its own name and ABN, even though it is not registered for GST?

The ATO has confirmed that an agent, acting on behalf of a supplier_id that is registered for GST, can issue a valid tax invoice with its own name and ABN on the invoice rather than the supplier_id's, even if the agent is unregistered for GST.

That is, there is no requirement that the agent be registered for GST.