What is "Court Games"?
Court Games is a quest that was added with the Summer season update and is given to you by Kirill.
Why can't I unlock the second part after completing both missions?
This particular quest is intended to be unlocked in increments, you will be able to unlock the next stages of this story after completing other quests in the Elysium. It's a side part of a bigger story!
However, please keep in mind that some of those quests may still be locked and will unlock at a later period, thus, the second part of Court Games will not be available for unlocking even when the player has completed all other currently available quests.