Converting Boolean circuit to Boolean formula in parallel
Take a look at the Tseitin transform.  The Tseitin transform can be applied in parallel, working separately on each gate.  Consequently, this can be implemented using one processor per gate, $O(1)$ time per gate, and the resulting formula has size $O(m)$.  Here I'm not counting the time to build the final formula; the final formula is a conjunction of the clauses produced by the processors, with each processor adding $O(1)$ clauses to the final conjunction.