Carried Interest Fairness Act of 2021 This bill modifies the tax treatment of carried interest, which is compensation that is typically received by a partner of a private equity or hedge fund and is based on a share of the fund's profits. (Under current law, carried interest is taxed as investment income rather than at ordinary income tax rates.) This bill includes provisions that set forth a special rule for the inclusion in gross income of partnership interests transferred in connection with the performance of services, treat as ordinary income the net capital gain with respect to an investment services partnership interest except to the extent such gain is attributable to a partner's qualified capital interest, exempt income from investment services partnership interests from treatment as qualifying income of a publicly traded partnership, exempt certain family partnerships from the application of this bill, increase the penalty for underpayments of tax resulting from failure to treat income from an investment services partnership interest as ordinary income, and include income and loss from an investment services partnership interest for purposes of determining net earnings from self-employment and applicable self-employment taxes. The bill defines investment services partnership interest as any interest in a partnership held by a person who provides services to a partnership by (1) advising the partnership about investing in, purchasing, or selling specified assets; (2) managing, acquiring, or disposing of specified assets; or (3) arranging financing with respect to acquiring specified assets.