I disagree. The wording breaks down to A or B, and C. That comma indicates that it's one of the first two clauses AND the third clause. Of course, that's the problem with using this kind of language and not having it broken into sections so that it's clear what's meant.

Realistically, this is going to be used to selectively enforce yet another BS law that disproportionately affects the lower socioeconomic strata. Look like you might be <insert whatever exploitable demographic officer dickhead picks today>, "I'm sorry ma'am, your pet isn't be appropriately supervised by anyone other than the driver."