Get a room you two!

Two numbers are chosen randomly, both are positive integers smaller than 100. Sharon is told the sum of the numbers, while Parker is told the product of the numbers. Then, this dialog occurs between Sharon and Parker:

Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I don't know the numbers.
Sharon: I don't know the numbers.
Parker: I do know the numbers.

What is the product of the two numbers?

Adapted from a certain post I've found on the internet.
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