Truss tweeted about the meeting with Parker, saying: "I want New Zealand to be one of the first free trade agreements the UK signs as we prepare to become an independent trading nation once again."
Parker said New Zealand and Britain had deep and long-standing ties built on a shared history and shared values.
"That, and our track record as an honest negotiating partner means it is natural we should be an early choice for the UK to form an even closer trade partnership – through the negotiation of a high quality, comprehensive and inclusive FTA – shortly after its departure from the EU."
Truss last week set out her three immediate priorities:
• Securing new trade deals – the top priority being an FTA with the United States;
• Establishing new free ports across the UK and attracting investment;
• Getting British business ready to trade for October 31.
Johnson wants amendments to the withdrawal agreement that May negotiated with the European Union but he has pledged to take Britain out by October 31, with or without a new deal.
Parliament has already voted against a no-deal and without the backing of Parliament, a hard exit could precipitate a no-confidence vote and early election.
If Johnson's Government survives, its primary goal will be to get a trade deal with the EU.
Former leading New Zealand trade negotiator Crawford Falconer is chief trade negotiation adviser and second permanent secretary for Truss' Department for International Trade.
New Zealand is at present negotiating a free trade agreement with the EU.
Foreign Minister Winston Peters has also been pushing the Donald Trump Administration in the United States to begin free trade talks.
He made the case in a speech in Washington this month on the sidelines of a conference he attended on religious freedom and in meetings with Vice-President Mike Pence and Secretary of State Mike Pompeo.
Peters told reporters at Parliament earlier this week that he came away encouraged about a free trade deal and that New Zealand was closer to a deal.
"Things were very, very positive about looking at New Zealand with new eyes and with a new sense of obligation, so I came away very encouraged. "
"For the first time for a long time, a number of countries are realising how we must all refresh and reset our views to secure economically and socially the security of our part of the world, the biggest blue continent, as we call it, worldwide.
He could not give a start date but said New Zealand was "a whole lot closer than we once were".
