Take a look at the sentence again:
On vacation, Patrick can trust neither
his roommates nor
the veterinary staff with Squeeze, his 9-foot
albino python, so he hides his beloved reptile
in a suitcase and sneaks him into hotel
rooms.
When you use neither ... nor, you must have two equal grammatical items. After neither, notice that you have a noun phrase, his roommates. And after nor, you have a second noun phrase, the veterinary staff.
In addition, the sentence lists two actions, hides and sneaks, but both are simple present tense verbs, which maintains the parallel structure.