Counterexample for simple parity automaton reduction
After some thought, I found a counter example. It has four states and there might be one with only three states, but this one works too.



States 1, 2, and 3 are language equivalent. 1 and 3 also have the same color, so they can be merged. If we merge 3 into 1, this is the result:



In the original automaton, $(aab)^\omega$ has the run $(103)^\omega$ which only sees color 1. In the reduced automaton, $(aab)^\omega$ has the run $101(201)^\omega$, which also visits color 0 infinitely often.