And a significant part of the Canadian constitution is now codified and entrenched in ways that no single one of the federal or provincial parliaments across Canada can freely amend, albeit not in a single document, even though Canada shares the same King as the UK. No reason the UK couldn’t do it - the UK Parliament itself even enacted the fundamental constitutional structure that Canada now has, at Canada’s request, and in the same act removed its own power to legislate for Canada going forward.
(Canada had previously deferred its assumption of the power to amend its own constitution without asking the UK to do it until it figured out what replacement arrangement it wanted, which took half a century and the requesting Canadian government still very controversially did not win the assent of or even consult Quebec before proceeding.)
With that said, there is an important structural difference: Canada is a true federal state rather than a unitary one like the UK which merely has some nonexclusive and constrained devolution to three subordinate parliaments within specific scopes. Every single bit of the Canadian constitution is indeed freely amendable by enough of the eleven Canadian federal or provincial parliaments working together. Certain specific parts can indeed be amended unilaterally by one parliament, but many parts need a much larger level of consensus, up to and including unanimity.
This means that the Canadian situation is not really a counterexample to the claim that the UK parliament would necessarily retain full amendment rights if it did codify a constitution, since the UK parliament is most similar in authority not to the Canadian federal parliament but to all eleven federal or provincial Canadian parliaments combined, which collectively do retain full amendment flexibility if they can all agree as required.
However, some provinces refuse to ratify amendments without a referendum, and the country has a lot of trauma from past failed attempts to make major constitutional amendments such that they mostly don’t attempt them any more, so the eleven parliaments have de facto lost some of their collective parliamentary supremacy even if they have not lost it de jure.