Pedantic comment: you don't need the previous results to compute the nth fibonacci number; the recurrence has a closed form solution. It is not constant time to compute (unless you assume constant time, infinite precision arithmetic) in practice, but it can certainly be done without the prior terms.

The rest of this post is on point, though. Serialization bottlenecks (Amdahl's Law) prevent a lot of optimizations from making meaningful dents in real workloads.