I have never seen a Parker Bros. or a Remington/Parker catalog offering any other single triggers. There was, posted on here some time back a Walter King letter to a customer where he states that Parker Bros. had "removed their Infallible single trigger and replaced it with double triggers."
The Infallible Single Trigger was the design of Alfred Harvey Worrest, and was made and installed by his Lancaster Arms Company in Lancaster, PA.
While I've never seen an Ithaca Gun Co. catalog offering the Infallible single trigger, they advertised such in 1915.
1915 ad for Flues with Infallible Single Trigger.jpg