There is definitely overlap between multithreading and parallel coding/computing, with the main differences in the target processing architecture.
Multithreading has been used to exploit the benefits of concurrency within a single process on a single CPU with shared memory. Running the same programs on a machine with multiple CPUs may result in significant speedup, but is often a bonus rather than intended (until recently). Many OSes have threading models (e.g. pthreads), which benefit from but do not require multiple CPUs.
Multiprocessing is the standard model for parallel programming targeting multiple CPUs, from early SMP machines with many CPUs on a big machine, then to cluster computing across many machines, and now back to many CPUs/cores on a single computer. MPI is a standard that can work across many different architectures.
Of course, one can program a parallel design using threads with language frameworks like OpenMP. I've heard of multicomponent GUIs/applications that rely on separate processing that could theoretically run anywhere. Practically, there's more of the former than the latter.
Probably the main distinction is when the program runs across multiple machines, where it's not practical to use multithreading, and existing applications that share memory will not work.