The joke uses the same techniques that zeugma does, but I wouldn't formally call this a zeugma or syllepsis.
First the difference between zeugma and syllepsis: syllepsis is a particular form of zeugma as zleptan noted. Zeugma is an ellipsis with parallelism in general; syllepsis is a zeugma where the parallelism involves amphiboly (multiple senses) in the common word, switching grammar or semantics.
For general zeugma:
As Virgil guided Dante through Inferno, the Sibyl Aeneas Avernus (Roger D. Scott, from Silva Rhetoricae, zeugma )
the parallelism is Virgil to Sibyl, Dante to Aeneas and Inferno to Avernus, eliding 'guide' and 'through'.
You held your breath and the door for me (Alannis Morisette Silva Rhetoricae, syllepsis )
'breath' and 'door' are parallel and 'held' is elided, but 'held means two slightly different things in the parallel unelided statements.
Now to the original question. In the joke, the question is ambiguous (which of the subject or object is being asked about) and parallel (only 'the snake' is used, not the verb or other noun). The response resolves the (unexpected) ambiguity.
I would call this syllepsis, because 'snake' is ambiguous as either the subject or the direct object.
So yes, it really can be said to be zeugma (and more specifically syllepsis).