Can private pilots drop skydivers if the pilot pays a pro-rata share?
This is not so difficult. Many people like to refer to the regs without actually looking at the regs. This is all taken from §61.113.
Pro rata share is not the same as getting compensated.

(c) A private pilot may not pay less than the pro rata share of the operating expenses of a flight with passengers, provided the expenses involve only fuel, oil, airport expenditures, or rental fees.

In this scenario, the pilot isn’t holding out as a commercial operator; these are his friends.
According to any online legal dictionary, the pro rata share is defined as the total cost divided by the number of shares. In other words, the total cost of fuel, oil, airport costs, and rental fees split evenly between everyone in the plane, including the pilot. As long as it is split evenly, it’s not compensation or hire, it’s cost-sharing.
All that said, NEVER do anything because some guy on stack exchange told you that it was ok. Check the regs for yourself and contact your local FSDO for clarification.