Do "sparrow grass" and "alligator pear" have any currency in spoken AE, or are these terms chiefly dialectal?
I've heard Alton Brown and others use the term alligator pear to describe an avocado. But, it was an attempt to sound campy rather than common usage.
I don't think anyone unfamiliar with this rare usage would immediately know your meaning at first blush.
As to sparrow grass… this is the first I've heard it. It is probably a folksy mispronunciation of Asparagus. Similar to old-timer's disease in place of Alzheimer's Disease.
Purely coincidentally, the word avocado comes to English by way of Spanish through serial mispronunciation. Derived from the term aguacate (avocado tree). This was itself a mispronunciation of the Nahuatl word āhuacatl (meaning testicle). Many incorrectly assume it comes from abogado which means lawyer (not only because of the similar pronunciation, but also due to the similar meanings of lawyer and testicle). Source Wikipedia