(updated based on Pieter's comments below)
There are (essentially) two sets of UTXOs at play. One is constructed from the blockchain itself, by reading the blocks on disk and updating with any new blocks that are broadcasted. This set is saved to disk, and can be reconstructed from the blockchain data if erased.
Another UTXO set is the mempool itself. Each output in the mempool is considered a valid UTXO, unless it is already spent by another mempool tx. Moreover, any outputs consumed by mempool transactions from confirmed transactions are considered spent in the other UTXO set. This mempool UTXO set is not saved to disk, and only exists when there are transactions in the mempool, while the node is running.
You can see existing CPFP transactions by calling
getrawmempool with the verbose option. This produces a json array in which you can see if an unconfirmed tx depends on an existing unconfirmed tx: