Some basics:
1) There never was a referendum on this issue*. There was a non-binding advisory plebiscite, by which it was made clear that the country was almost exactly evenly split on the very vague question of 'Should the UK leave the EU, or remain in it'.
2) The sovereign powers in the UK lie with parliament. Not with the people; Not with the government; Not with the Prime Minister. This was resolved by means of a civil war in the 1640s.
3) The purpose of the 2016 plebiscite was entirely party political, and had fuck all to do with anything other than keeping the Conservative Party rank and file in line behind the then Prime Minister. That it was ever allowed to affect the entire nation is a disgrace; That the current PM has used it to try to establish herself as the holder of sovereign power in the nation (in the name of enacting the 'will of the people') is a crime. The people have an opinion, not a will; That opinion is 'we don't know'; And 'the people' are not in charge here - that's not how the UK is governed, and never has been.
There's no question that brexit is hugely harmful to the nation. Parliament recognises this fact. Parliament therefore has a duty to stop this harm from befalling the nation. They also have the (sole) authority to impose their opinion, and need to stand up to May, who seems to be labouring under the misapprehension that the UK has a President, and that she holds that office.
Brexit is currently being used as an excuse for a coup attempt by the government, to overthrow parliament, and elevate the PM to an autocratic position. Parliament must not tolerate this any longer.
The UK doesn't have direct democracy. If Ms May, or Mr Farage, or anyone else, wants to change that fact, then they need to introduce a bill to that effect in parliament, and to gain sufficient support in both chambers to pass that change to the UK constitution.
Nobody, not government, not the Prime Minister, and not the people of the UK, has the right to make any demands of any kind on the basis of the results of the 2016 plebiscite. That may be something that many feel is wrong, even immoral. It may even actually be anti-democratic, for many definitions of 'democratic'. But that's how the UK is governed, and to attempt to circumvent that structure in which parliament is sovereign are literally treasonous. Regardless of the half-baked opinions of the owners of the tabloid press (and their moronic readership).
Other than the 1975 referendum, which was an actual referendum, in which the people voted overwhelmingly (67 - 33%) to remain in the EEC. Binding referenda like the 1975 poll are rare, but such 'real' referenda should ideally be required before any very long term or fundamental change to the structure of the nation. A binding referendum on a fully detailed change to the law should, perhaps be held in the near future, as one way of resolving the current crisis. But such a poll would require a fully detailed proposal or bill on which to vote.