Not every pen is fit for a king, as can be seen in this article from The Guardian: ‘Oh god I hate this’: King Charles expresses frustration over leaking pen
His mother seems to have had more luck with pens. In recent days it wasn’t uncommon to see footage showing Queen Elizabeth II signing documents. This often included footage showing her using what seems to be the Parker 51.
Not only her pen, also her ink was probably from Parker, as described in this blog post that was posted here in 2018: Stationery Factlets #3: The Queen’s Ink