Is the word problem of CFLs in NC?
The problem is not in NC unless P=NC. Jones and Laaser showed in "Complete Problems for Deterministic Polynomial Time" that the problem (therein called CFMEMBER) is P-complete.
Interestingly, Ullman and Van Gelder showed in "Parallel Complexity of of logical query programs" that the same problem where the input CFG has no epsilon productions is in NC (Corollary 7.2).