Can all unambiguous grammars be parsed in linear time?
Unambiguous context-free parsing is in $O(n^2)$ using Earley's algorithm.  Whether there exists a parsing algorithm working in linear-time on all the unambiguous context-free grammars is an open problem.  One of the most advanced statements of this kind is due to Leo [1991], who showed that a variant of Earley parsing works in linear time for all LRR grammars.

[Leo 1991] Joop M. I. M. Leo.  A general context-free parsing algorithm running in linear time on every LR($k$) grammar without using lookahead, Theoretical Computer Science 82(1):165--176. doi: 10.1016/0304-3975(91)90180-A