Random Access vs. Parallelism modes of operation
Random access (read, or write, or both) is a property of a mode of operation. Parallelism is a form of computation where multiple independent execution units can work together to complete an operation faster. They're somewhat orthogonal concepts, although random access does imply that a mode of operation is trivially parallelizable.

Random access is always a good thing because it allows more flexibility in how the mode of operation can be used (for instance, random access modes are desirable in full-disk encryption because you don't want to be waiting for your entire disk to be decrypted just to read a small text file on it). There are perfectly secure modes of operation available with either non-random-access and random-access. Random-access modes do generally tend to require a little more care with nonce/IV selection than modes without due to the constructions that tend to come up when designing for random access, but there is absolutely no reason to equate random access capability with security.

Random access has nothing to do with security; if a mode of operation that allows random access is insecure, then it is a bad mode of operation and should never be used, regardless of its random access properties. Whether an attacker does or does not leverage CBC's random access properties when decrypting has no impact on CBC's security properties (stated differently: CBC is what it is; the fact that it happens to allow random access during decryption has no bearing on its security properties).