The correct answer is that parliament itself called the referendum and effectively deferred the decision in principle to the public. It cannot claw that decision back because it didnt like the answer. It must, however, find a way of implementing it that it defines as being in the public interest.
One can remain sovereign whilst delegating individual decisions.
It remains, however, a decision in principle only as nobody asked the public to vote on different types or aspects of leaving. Parliament retains the authority to make those decisions.
Its messy but its constitutionally acceptable. The only problem is...well, parliament is a bit **** at this sort of thing.