Although he initially described the reports of the offer as "supposedly leaked" he later confirmed the reports were an accurate reflection of what was being offered, but added that the offer was unacceptable.
"This latest offer by the European Union reflects agricultural protectionism in the European Union," Parker said.
"It is a very negative signal, especially at a time when we're working together in the World Trade Organisation and elsewhere encouraging countries not to be protectionist and ensure trade can flow freely between us."
Parker said in 2019 the EU exported the equivalent of just under 1kg of cheese to New Zealand for every New Zealander, tariff free.
In contrast, the offer the EU had put on the table would allow New Zealand to export the equivalent of about three grams of cheese for every European Union citizen annually and "even then it's subject to volume restrictions and highly prohibitive tariffs," Parker said.
New Zealand has been arguing for a free trade deal with the European Union - the world's largest trading bloc - for more than a decade. Formal talks began in mid-2018.
After his speech, Parker said he would speak to the EU trade commissioner, Phil Hogan, on Monday evening (NZ time) "to register very clearly my dissatisfaction and disappointment with this offer as well as the fact that it's a leak."
While he refused to describe the offer as a sign the EU was not negotiating in good faith, Parker said it did raise the question of whether Europe wanted a deal with New Zealand.
"The EU said they wanted to finish their negotiations by the end of July. It's hard to see how they could do so if that's their pitch," Parker said.
"We just can't land this in this place and they will have to reflect on whether they really want a trade agreement with New Zealand.
"We want one with them. We think it's important that liberal democracies around the world co-operate. We've been back of the queue for a long, long time, and it's disappointing that this far into the negotiations their first goods offer in respect of these sensitive issues is so low."