This is not a problem in ETH [1]. It's a bug in one of the contracts written by Parity. Anyone that has (had) money in these specific contracts is in trouble. If you are not using these specific contracts, you are safe (for now).
ETH and ETC have the same underlying technology and language to write the contracts. So if someone writes a contract it can be used in both variants. The contract may check that it is in ETH or ETC (probably looking for the contract that was added during the fork) but I think very few contracts check this, so almost all contracts are usable in both chains.
So if it was possible to use the Parity contract in ETH, then it's very probable that it was also possible to use it in ETC.
To answer your question: Probably the same bug that was used to steal ETH can be exploited to steal ETC, if someone is using these contracts in the ETC chain. But it's not a problem in ETH or ETC [1].
[1] As other commenters noted, it's a very bad design flaw to make all function public by default. It's not an error, but it makes much easier to write buggy code.
