How would you recommend parsing a private bitcoin core node with nodeJS?
This question is pretty broad as parsing a node involves decoding every transaction within a block(which would require txindex to be enabled), and recording each output and input until you end up with a list of all outputs that have not been used as an input aka an unspent output.   

With a fully synced node and txindex=1 you can begin iterating through each block of the last 200 blocks with loops using the rpc calls getblockhash getblock getrawtransaction and decoderawtransaction, storing the outputs in a database. On each new block you should check every input to see if there is a transaction id matching an output in your existing output database, if there is that output has been spent. The node-bitcoin JSON-RPC wrapper should work fine for implementing these calls to your node. As you can imagine this would take some time.