Why do 'fine words butter no parsnips'?
According to this article, the term is less about the parsnips and more about the butter. An alternate variant is "fine words butter no fish". I think the expression is contrasting the conversational effects of flattery, empty promises, etc. (cf. "to butter up") with their lack of practical utility.
The association of parsnips with butter is of long standing. Fletcher alluded to it in his play ‘Womans Prize’ in 1625. The OED’s earliest citation for the acttual proverb is this from 1639: ‘Faire words butter noe parsnips, verba non alunt familiam.’