CBC in its standard form is not parallelizable at all. However, like for hashes, one could decompose the data in several parallel streams that are encrypted independently with CBC, for multiplying the speed.

CBC with a block cipher has been completely obsolete for decades, even if the huge inertia of cryptography software development has kept it in use in many places, even until today.

However, CBC has a descendant that is still useful. CBC is a derivative of CFB, from which it differs only in the place from which the output is extracted from the encryption automaton.

There is another derivative of CFB, where the block cipher function with a secret key is replaced by a wider bijective mixing function, where the output is extracted by truncation. Here the role of the secret key is played by the part of the state that is not extracted in the output and the encryption automaton obtained thus is equivalent with a CFB mode for a block cipher where the key would be changed after each block. The fact that no 2 blocks are encrypted with the same key eliminates all security problems of the traditional CBC/CFB.

Only the fact that encryption and decryption are not parallelizable is a disadvantage, but this is partially mitigated because integrity verification is a byproduct that is provided without additional expenses (by extending the string to be encrypted with a block of known content).

This mode of operation is what the team Guido Bertoni, Joan Daemen, Gilles Van Assche and Michaël Peeters has called "duplex encryption for sponge functions".

While that team has contributed many important things to the progress of cryptography, I hate that they have always coined a great number of new names for old things, and always without mentioning at all the correspondence between their new names and the old concepts that had been used for decades before them.

Their initial contributions from 2007 to 2012 were all about what is obtained when in all those of the classic modes of operation that were known by 1980 in which the inverse of the block cipher is not needed, one substitutes the block cipher with a truncated wide mixing function (in the Shannon sense), but for all of those they have invented new fancy names, like "sponge absorbing" (derived from CFB-MAC), "sponge squeezing" (derived from OFB), "donkeySponge" (derived from the Alred optimization of CBC-MAC) and the like.