I'm just syncing Parity wallet and am wondering if I need to do something to enable the wallet to accept ERC20 tokens, such as EOS. Or do the tokens simply show up one they've been deposited to an Ethereum address.
The simple answer is: tokens simply show up once they've been deposited to an Ethereum address.
Yet, you should understand that actually tokens are not "deposited to Ethereum address". Token balance is the amount of tokens associated with Ethereum address by some Smart Contract. In order to find out a full tokens' balance Parity has to ask all known Smart Contracts about the balance of particular address. That is why Parity only displays tokens which are registered with the token registry dapp (according to this github issue).
In general you should not care about this since all major tokens based on Ethereum should be shown in Parity without any additional actions required from user.
If you'd like to register your own token you might be interested in this page from official Parity documentation