The issue of Parliamentary Sovereignty arises in cases where there is a conflict between different Sources of Law - which law should prevail?
Because of the way the Referendum Case was presented in the High Court, much of the High Court Decision was devoted to a presumed conflict between the Sovereignty of Parliament and the Royal Prerogative.
But there is no such conflict in the Real World Case.
Referendum Act 2015 was enacted by Parliament. And the Point at Issue is not a conflict between the Parliament and the Executive, but the scope of duties and powers the Parliament has bestowed upon the Executive by enacting Referendum Act 2015. This is not an issue of Parliamentary Sovereignty, but of Interpretation of Statutes.
If to accept that Referendum Act 2015 is binding on the UK Government, then this act lays on the Government the Duty to perform all the necessary steps to terminate the UK membership of the European Union.
The only necessary step of the Leave Procedure that the Government needs to take to terminate the UK membership of the EU is to formally notify the EU of its intention to leave the EU.
Because the UK membership of the EU gives rise to various Real World relationships between the affected persons (a company is a legal person), it is also necessary to give to these affected legal persons reasonable time to terminate these relationships.
Such time to terminate relationships is normally expressly stated in contractual agreements within a termination clause.
If such time period is not expressly stated, the parties are presumed to be obliged to give, and entitled to be given, such "notice period".
For example, according to Natural Justice (Common Law): a person employed at a weekly wage, is entitled to a minimum weekly notice (or payment in lieu), even if there was no written agreement.
Thus, the only action required from the UK to terminate its membership of the EU is a formal written notice (with a specified period of time to terminate current relationships, if such time had not been provided in the termination clause of the EU Treaty).
And this obligation and powers to terminate the membership of the EU is the only Royal Prerogative logically following from Referendum Act 2015.
RA 2015 does not grant the Executive any prerogative to make any agreements with the EU about any future relationships after the UK terminates its membership of the EU. Any such agreements are not part of the Leave Procedure, and have nothing to do with the RA 2015. The prerogative to make such agreements should either exist (like making foreign treaties not affecting domestic laws), or be granted by a specific Act of Parliament.
Thus, if to assume that the referendum under Referendum Act 2015 is binding, rather than a "General Public Consultation" (opinion poll), then the UK Government has the duty and powers to terminate the Membership of the EU, without need for any further authorisations by Parliament by formally notifying the European Council of its decision to leave the EU.
But, if the UK Government wants to negotiate any agreement about the future relations with the EU, and this agreement will have effect on the UK Domestic Law, then it would need an Act of Parliament to authorise such agreement.
Thus the conclusion, based on the assumption that the Referendum was binding on the Government, is that the UK Government can proceed to notify the EU about the People's Decision to leave the EU without any involvement of the Parliament. The Parliament has already authorised the Government to do so by RA 2015.