The pro-EU faction want a parliamentary vote as a desparate last chance to stop BREXIT - whereas HMG Parliament already have the sovereign right to honor the declared wishes of the British people without further voting.
The government famously, and at great taxpayer expense, sent a referendum pamphlet to everyone in the UK declaring that:
"This is your decision. The government will implement what you decide".
But when we decided for Brexit, what happened? The pro-EU faction decided to attempt the same betrayal that they succeeded in in Ireland and France.
Those countries were betrayed and cheated out of their anti-EU decision. It's foolish therefore not to see the push for a second vote, this time in Parliament, as anything but a cynical ploy.
The referendum vote did indeed determine that Brexit would happen: but it gave no instructions on how it should happen and which of the many possible interpretations of Brexit should be pursued. Only Parliament can do that.
Inserting a device of direct democracy (a referendum) into a system based on representative democracy is always going to be messy, which is one reason we have so few of them. But it would be less messy, and the Parliamentary consequences more straightforward, if Parliament had committed itself to being bound by the outcome in the first place. But it didn't, we are where we are, and given the government's apparent determination (since somewhat tempered) to ride solo, something like the High Court/Supreme Court case was inevitable, never mind who brought the case or what they were ultimately after.