the waiter to Hank’s party

[sitifan]

“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?

[Raymott]

“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?

"Hank's party" is the group of people that Hank is with. See a dictionary for definitions of 'party'.
(I assume that the waiter is at a restaurant, not at a party).

[emsr2d2]

“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?

"Please wait a moment" said the waiter.
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).

[billmcd]

“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?

The waiter is addressing/directing comments to a group of people (Hank's party). "Party" in your example is described/defined previously by emsr2d2 & Raymott.