Partial collisions of hashes, why is this important? There's a lot of work finding "partial collisions" in hashes, and "Parallel Collision Search with Cryptanalytic Applications" seems to be the go-to paper for this work.  I'm trying to understand why finding partial collisions, particularly by brute force, is such a topic of interest. As an example, let's say we have a 256-bit hash, and the last 4 bytes are the same for two message.  

Is the interest in this work just because it's a useful segue to understand how to make collision?  Is there a condition where one would truncate a longer hash that would cause a partial collision to be a problem?
This seems to boil down to whether or not anyone would have a reason to truncate a hash. The answer is yes, absolutely. For example, someone who needs a 256-bit digest may want to use SHA-512/256 instead of SHA-256 on a 64-bit processor for improved performance (the former uses 64-bit operations whereas the latter uses 32-bit operations). SHA-512/256 is nothing more than regular SHA-512 with a distinct IV, truncated to a 256-bit value. Likewise if someone wants a hash with 280 collision resistance, they would want to truncate a stronger hash like SHA-256 to 160 bits instead of using SHA-1 which has cryptographic weaknesses that allow collisions to be found faster than predicted by the birthday bound.

Another reason someone might want to truncate a Merkle–Damgård hash function is to resist length extension attacks. SHA-256 is vulnerable to them, whereas SHA-512/256 is not, despite being of the same size. An efficient collision against such a truncated hash could then be considered problematic.