Would you treat the contralateral neck in post-op parotid salivary gland malignancy with multiple positive nodes ipsilaterally?

Not unless there are suspicious contralateral nodes.

Parotid drainage is ipsilateral and it makes no clinical sense to me to treat CL neck electively.

Of note, the paper cited is on submandibular gland cancers, not parotid. With SMG, there is a chance of CL mets if the initial presentation has very anterior tumor at the edge of IB vs 1A. However, it's unclear how carefully that can be ascertained via this retrospective study. It would not be a good idea to extrapolate this to parotids.

No, unless there is a prior incision (even a remote one) that could shunt drainage to the contralateral neck.

The parotid gland is a lateralized structure and drains to the ipsilateral neck. In addition to the noted differences with the cited Hsieh study of submandibular cancer, I would add that Stennert et al., PMID 12874071 reviewed 160 major salivary gland cancers (149 of which were parotid cancers, only 11 were submandibular) and found no contralateral lymph node metastases.