Are self-stabilizing algorithms guaranteed to work in parallel?
I suggest you look for a textbook that provides an introduction to self-stabilizing algorithms (with some definitions, examples, etc.), and work through that textbook.  I think that will provide you a much better understanding, and you'll be able to answer this kind of question yourself.  Trying to learn a subject by reading Wikipedia and asking questions on StackExchange sites is not the way to go.

For instance, I think you'll find that, yes, of course each distributed node is working concurrently (in parallel); that's what it means for a system to be a distributed system.   And yes, for a self-stabilizing algorithm to be correct, one must show that it will never enter an infinite loop of the sort you describe; this is the convergence property.

Designing an algorithm with these properties is non-trivial, but in general, designing self-stabilizing systems is not easy.  No one ever promised this stuff would be easy.