Legally, the King can exercise the royal prerogative unilaterally. However, it’s widely thought that if he ever did so, Parliament would act quickly to remove the last of that power, so it’s exercised on the advice of the Prime Minister as a matter of convention.

ETA: Apparently, it’s debatable whether this also applies to dissolving Parliament, even though that’s a prerogative power. See this article on the Royal Prerogative in the UK