Seems easy enough to use a parallel scan if you're willing to accept a little work inefficiency, right? Assign each scanner thread a block, first each one counts/xors how many quotes are in its block, exclusive scan on those (last thread's result is unused), and you have the quoting state at the start of each block. And hopefully that block's still in the core's cache.

Or since newlines in strings should be rare, maybe it works to save the index of every newline and tag it with the parity of preceding quotes in the block. Then you get the true parity once each thread's finished its block and filter with that, which is faster than going back over the block unless there were tons of newlines.