Those are "opportunity funds". "Parallel Funds" normally are funds that invest at the same time and in the same fund percentage as the main fund. These are often setup as a way to group together LPs; for example you may have a main US fund and one in the Caymans that's for foreign investors. They function as a "Fund Family" and because they make the same investments at the same percentages, they normally have the same returns.
Looking at the SEC filing for Pinterest, you can see this is the case for a16z. Fund III and Parallel Fund III invested in the same rounds, always at the same proportion:
Looking at the SEC filing for Pinterest, you can see this is the case for a16z. Fund III and Parallel Fund III invested in the same rounds, always at the same proportion:
https://www.sec.gov/Archives/edgar/data/1503674/000114420419...
What's not clear given this, is why the parallel fund has much better returns. It may have to do with how the fees are structured.