That is the lay out of the Parker SST, trigger in the righthand slot and selector in the lefthand slot, but that is not the normal Parker thumbpiece on the selector.
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I don't have any experience with Parker SSTs, but I have a 16-gauge Fox-Sterlingworth Skeet & Upland Game Gun that had a problematic Fox-Kautzky SST. Worked fine as long as it was selected right/left. Not so if selected left/right. Maybe I should have left it selected right/left, but for $300 Keith Kearcher made it double trigger.