"So yes, I expect there will be a few sensitivities there and then a few other issues that are sensitive for both sides, but again, we are well aware of them. There will be no hidden agendas here, we are very open about our sensitivities and we will try to find a good compromise."
Another area of contention could be around geographical indicators.
The EU has "protected geographical status" for a range of goods, such as Brie cheese, meaning those names can only be used on cheeses from the areas they are named for.
Parker said Malmstrom had asked negotiators to work through those complicated areas early on in the process so they did not cause delays at the end.
"I think that demonstrates a willingness on the part of the European side of the negotiation, which we share, to bring this to a conclusion as soon as we can."
Parker said the EU and New Zealand were broadly aligned on the issue of patent protection for new medicines and did not expect that to be as much of an issue as it had been dealing with the US prior to their withdrawal from the TPP negotiations.
Two-way trade between the countries was currently worth $21.5 billion and $16b without the United Kingdom, which will leave the EU from next March.
Malmstrom also referred to the tariff war involving the United States and China as a major concern, referring to New Zealand as a "strong partner" for the EU in trying to hold up the World Trade Organisation rules.
She said the EU was worried about escalation of that, saying it would affect companies and consumers worldwide if it escalated.
"So that is a concern and we are trying to work with partners, and New Zealand is a very strong partner in that, to see how we can strengthen and develop and elaborate new rules within the WTO because it has served us well. It isn't perfect, of course, but it has served us well."
The EU has said it will go to the WTO to challenge the United States' tariffs on steel and aluminium and has threatened retaliatory tariffs.
She said one area in which WTO rules needed modernising was to reflect the increasing importance of sustainability, the environment and labour rights in trade agreements.
"We are very eager our agreements include sustainable development, respect for the environment, for labour rights. And there the WTO is not up to it so we need to set new standards.
"I think we can lead the way and do a blueprint here that can inspire others."
Parker said the agreement could not only work for New Zealand but act as a guide for how other trade agreements should look.