Isn't that true of any such system? The US may have a written constitution, but that would be just a piece of paper if it weren't for the conventions surrounding it.Perhaps, but I think there is a profound difference in the legitimacy and authority of a system of government if there has at some point been a clear demonstration that the general public consent to it. The ideal might be a popular vote to accept the original constitution document, with a substantial majority in favour, and then some practical mechanism that allows any member of the public to cause a later change to the constitution if the consensus of their peers agrees with them. The important point is that the final authority on the system of government is the people being governed, directly.
In practice I'm sure we could find a way to restrict the authority of parliament - the judiciary have already signalled some willingness to question the validity of laws if they undermine the integrity of parliament as a democratic body or basic human rights.
According to our current constitution as generally accepted, the courts can't overrule the will of Parliament. That is parliamentary supremacy in action.
Human rights cases heard by our highest courts have typically come down to favouring an interpretation of statute law that is compatible with other statute law like the HRA in cases of ambiguity. It is understood that if Parliament clearly intends a new law to reduce the legal protection of human rights then our courts must follow that intent in their judgements.
After that, we're getting into some very fuzzy areas that come to the foreground with issues like Brexit or Scottish independence, but again the courts will enforce the will of Parliament (and by extension, the ability of Parliament to express its will).
I personally don't think either of those is necessarily a good thing, which is why I believe it could be better in the long term if we had a proper written constitution as many other states do, setting out the basic structure of our government and associated responsibilities, authorities and limits on power, with explicit popular consent to that form of government.