The price of a transaction is equal to gasUsed * gasPrice. gasUsed is ultimately set by the protocol, and would be the same on different forks. gasPrice is set by the user, but miners can choose not to accept a transaction if they think that the price is too low (or for any other reason).
As such, if the price of ETH doubles, you can halve the gasPrice and pay the same fee. If the price of ETH halves, you can double the gasPrice and pay the same fee. The cost to a miner of actually executing a transaction is the same, either way.
So to answer your question, it depends on whether ETC miners are willing to mine cheaper transactions, or whether ETH miners would demand more expensive transactions. Theoretically, neither has a reason to do so, aside from doing so accidentally leaving the gasPrice they ask to the unadjusted default. (Parity, as far as I know, compensates by changing the gasPrice relative to the current price of ETH in USD.) So there shouldn't be any real price difference between running on one or the other.