Yes, definitely the best introductory book on the subject I've ever seen. Indeed, the Sipser book is a model for how to write a very readable, accessible CS theory textbook.
As far as how complexity theory on parallel computing, communication complexity is one related approach:
Learning about NP complete problems is interesting to avoid certain pitfalls and mapping one problem to another is always a valuable technique, but it seems you are fairly new to analysis of algorithms so imho (having been a phd student focused on algorithms) this book and course is a great place to start.
If you are in the mood for something advanced, take a look at Introduction to Automata Theory, Languages, and Computation by Hopcroft, Motwani, and Ullman.
It has a good amount of proofs and a pretty strong focus on automata thought so that may not be your cup of tea.
Hopcroft won the Turing award in 1986 so he knows what he's talking about.
