“Please wait a moment,” said the waiter to Hank’s party.
What does "the waiter to Hank’s party" mean? Why not use the preposition at?
Who did the waiter say that to?
He said it to Hank's party.
As Raymott said, this is a different meaning of the word "party". It's not a celebration, it's a group of people.
I have booked a table at my favourite restaurant for a party of 8 = I have booked a table for 8 people (there will be a total of 8 people in my party).