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I think what the interviewer meant was distributed cache, much like memcached or coherence. In essence, distributed cache is internally implemented using Distributed Hash Table. While implementing DHT, we need to keep in mind about CAP theorem i.e Consistency, Availability and Partition tolerance. Only two out of this three can be fulfilled at one time and depending upon your requirements, you can pick any two. I

By parallel cache mean the cache memory shared by a number of cores (in processor).
As the L1 Cache is dedicated to seperate cores. L2 Cache is dedicated to 2 cores (in dual core processor) And is generally larger in space then L1 Cache the L2 Cache is jointly used by the 2 processors.Parallel Caching means sharing of cache memory to more than one processor.(Used to overall increase performance increasing cache hit latency....

In my experience with parallel programming, the major threat I found is false sharing. It occurs when your threads are modifying variables that reside in a single cache making it invalidate and do updates on all others. To ensure data consistency across multiple caches, multiprocessor-capable Intel® processors follow the MESI (Modified/Exclusive/Shared/Invalid) protocol. On first load of a cache line, the processor will mark the cache line as ‘Exclusive’ access. As long as the cache line is marked exclusive, subsequent loads are free to use the existing data in cache. If the processor sees the same cache line loaded by another processor on the bus, it marks the cache line with ‘Shared’ access. If the processor stores a cache line marked as ‘S’, the cache line is marked as ‘Modified’ and all other processors are sent an ‘Invalid’ cache line message. If the processor sees the same cache line which is now marked ‘M’ being accessed by another processor, the processor stores the cache line back to memory and marks its cache line as ‘Shared’. The other processor that is accessing the same cache line incurs a cache miss.

By parallel cache you mean multithreaded (on one computer) or muti-machine like memcached?