New Zealand and other nations whose exports are bound by quotas are concerned the split will give them less flexibility.
Previously it had been possible to send more to one of the EU nations when the market in another wasn't going so well but a separated out UK quota makes that more difficult.
"We want to be fair to those who are currently exporting to the UK, including New Zealand, and to be fair to our own producers and to be fair to our own consumers," Fox told Q&A on Sunday.
The UK would look at what was happening in its market over a range of commodities and with respect to a range of countries, he said.
"How much have they actually sold into the UK over the last three years? And even if that's a very high proportion of the EU quota, we will call that our baseline quota," he said.
He deflected a question as to whether the UK quota arrangement post-Brexit could cause a World Trade Organisation dispute, saying exporters would have to show they had been harmed.
"From the UK's point of view, it's the beginning of a two-stage process, because we will want to get our own established trading relationships with countries outside the EU when we leave.
"And so there's the basis for a further discussion," Fox said.
Parker has said both nations were committed to laying foundations for a UK- New Zealand trade agreement.
"We welcome Secretary Fox's visit as an opportunity to strengthen our close relationship at a time when the United Kingdom is seeking to reshape its relationships around the world following its decision to leave the EU," he said.
