The Privy Council is older than Parliament. It dates back to William the Conqueror, and was a council of advisors who helped him govern the state. It was nicknamed "privy council" because members of the council were privy to the thinking of the king.
After Parliament got created 800 years ago, the position of Lord President of the Privy Council is always held by the person who holds the position of Leader of the House of Commons (who always belongs to the governing party). That's because the job of governing the country has transferred to Parliament.
The person who currently holds that role, Penny Mordaunt, played a big role in the coronation, as she was representing Parliament and holding the Sword of State, which represents all authority and law in the land. (For this reason the Sword of State is also part of the ceremony in the state opening of parliament).
Normally the Lord President of the Privy Council would wear a black and gold uniform like the one below (this is what was worn at Queen Elizabeth's coronation):
But because Penny Mordaunt is a woman, she decided to have a caped dress made. But notice her dress and headress are decorated with the same gold fern embroidery as the original costume.