Hardest optimization problems in NC
Are you aware of the work on positive LPs/SDPs? There are a bunch of results in the area, mostly along the lines of "if the constraints of the LP/SDP are positive, then the problem can be solved in NC."

Some important references in this line of work are Luby-Nisan 93 and Jain-Yao 11. Another excellent resource is this slide of Rahul Jain's talk at the "Recent Progress in Quantum Algorithms" conference at IQC. The entire talk is available on youtube.