There are tons of examples of these sort of computers.
They have been around since, at least, the 1970's.
If you are "TRULY" seeking a situation where the processor ONLY performs a single thread, you likely will not find it. EVERY implementation I have seen requires the processor to perform additional tasks. (At least implementations where the system is TRULY a general purpose computing device. Communications to other processes, synchronization of datasets, results, etc.....)
Generally systems are, what you are terming, pseudo-parallel because it extends what most designers are used to. Initially processors were VERY expensive, VERY complicated for the day. It made sense to utilize a processor in as many tasks as possible, so time from the processor was allocated to a process/thread through a timeslice situation. (Definately a whole bunh of nuances as to what a "timeslice" is going to be.)
Now, as what was complicated is quickly becoming run of the mill, more and more systems are created from multiple processors each doing less, but doing it simultaneously.
This REALLY hurt Cray as while they built some of the FASTEST processors availabe they were really late in getting onto the multi-processor systems. (No, I am not talking about 4 or 16 processors or even 64, but rather higher numbers.)
Likely the FIRST company that did a really good job at getting multi-processors to do tasks well was Thinking Machines Inc when they made the Connection Machine.
Likely you, and many others on this board, have seen one of their systems. (Or at least a system that looks nearly, if not, excactly like one of theirs.) Wargames. The computer system that was shown as the one controlling the missiles is, is looks very much like, a Connection Machine.
This was a MASSIVELY parallel system that could have up to 16,384 processors. The LED's show communications accross the processors.
Steve Ciarcia from Circuit Cellar had a project some time ago that used multiple 8051 microcontrollers on a communications bus to solve mathmatical problems.
The Earth simulator is another example.
ALL of the Neural Net systems are essentially "parallel". (Actually in this case, they mostly do single tasks.) These are VERY simple processing elements that have the ability to run in parallel.
I think SETI@Home is another good example. (As previously mentioned.)
Look that the IBM the Blue Gene/L system. The one being built for Lawrence Livermore National Laboratory will contain 131,072 processors. (400 person staff, 1/2 a tennis court, 6+ megawatts of power, etc.... Truly amazing specs overall.)
Even in PC's there are various processes that are now handled by separate processors. The "KEYBOARD" is the oldest example as a separate procesor existed to perform ALL of the key scanning and debounce functions. Some diskcontrollers are separate processors, sound chips, video controllers. While these are NOT general purpose processors that can have their mission altered much, at least in the configuration being used, they are separate processors that perform tasks in parallel to the main processor.
Lastly, isn't this cheating?
You can easily look up more information on anything that I've written on the Internet.
The ones I picked are may favorite one, a hobby example, and a truly amazing current peice of work. (The Connection Machine, Circuit Cellar, and Blue Gene/L, respectively.)