What is this plant that sometimes accompanies parsley? I frequently buy bunches of flat-leaf parsley. Every now and then I find one or two stems of a different-looking plant included. Here is an image, with normal flat-leaf parsley on the left. I remember it having a grassy, plausibly parsley-like smell, but nothing strong.

Is this some variant of parsley, or a different plant altogether?
I believe your mystery plant is....parsley.  This question was asked and answered here.
It appears that the best explanation is a leaf variant on the common cultivar "Italian Giant".  Having grown this cultivar in my home garden, I can attest to having personal experience with this leaf shape.  Chop it up and use it as you would the rest of the parsley.