The doctrine
Parliamentary sovereignty is the principle that Parliament is the supreme legal authority, able to make or unmake any law, with no court able to override an Act of Parliament. The Victorian lawyer A.V. Dicey gave it its classic statement. Two things follow. No Parliament can bind its successors, and judges apply statutes rather than sitting in judgment on them.
Why it is the heart of the political constitution
Sovereignty puts the last word with people you can vote out. That is the essence of the political constitution, and the reason Starkey treats it as the load-bearing wall of the whole system. Take it away, and contested questions move to judges instead.
How it has been challenged
The doctrine has been tested more than once. During EU membership the Factortame case in 1991 let courts disapply an Act that clashed with EU law, a limit removed by Brexit. The Human Rights Act, devolution and the creation of a separate Supreme Court all redistributed power. Some judges have even hinted that sovereignty might have limits. On the Starkey account these are the cracks through which the legal constitution entered.