A contract can be deployed on different chains at the same address by using the same private key with the same nonce on each chain.
Given that forking a chain opens up possible replay attacks on acounts, does deploying parallel contracts also open up similar attacks upon the users of those contracts?
There is a mechanism to protect against replay attacks on different chains, it's the chainId param in EIP 155. Since you have to specify the chainId in your transaction params, the signed transaction becomes different for every chain, so no, nothing to worry about here.
The contract address is a hash generated from the wallet address that deploys it plus the nonce. That's why even if you deploy the same contract on multiple chains, the contract address will most likely be different.
https://ethfiddle.com/lOvRUBOJso - this is a simple fiddle, but if you click create multiple times, you'll see how the contract address changes each time.
That said, I think you have a misunderstanding on how contracts work. The person who calls the functions are the ones signing and paying for the transaction. So just because multiple contracts exist on different chains, doesn't mean that the private/public key pair of the wallet that deployed it is compromised.
