I generally think it's more liberals (in this America at least) that are likely to oppose zoos than conservatives as uch of the animal activists are liberal and much of the zoo activist movements have happened coastal US cities.
To an extent, but not necessarily. I've clicked on some profiles that have left virulent anti-zoo comments on facebook out of curiosity, and plenty of them are full-on MAGA, whereas I know many zoo aficionados that are comfortably to the left of Bernie Sanders.
You could go down quite a rabbit hole trying to answer the question of the politics of zoos. On one hand, conservatives tend to like zoos (in theory) because of the association of good old fashioned family outings, weekends with the whole family going to the zoo. On the other hand, they tend to be much more opposed to the use of tax dollars to fund things like zoos, and there have been many backlashes against what they've perceived as liberal bias in zoos - talking about conservation, climate change, etc. A few years back, some conservative mommy-blogger in Chicago did an "audit" of Brookfield Zoo (and Field Museum) looking for perceived liberal bias, Tulsa Zoo had the whole to-do about teaching evolution (the city wanted them to teach creationism as well) and the Ganesha statue, and Smithsonian National Zoo is subject to the new EO about liberal teachings (so we'll see if the salamander exhibit in the reptile house, which discusses coal mining and it's impact on species, survives the next few years).
Liberals, as you've noted, tend to be sensitive to animal welfare/rights issues, which in some cases can lead to opposition to zoos - it's no coincidence that one of the most perpetually beleaguered and underfunded major city zoos in the US is San Francisco, while NY and LA are constantly dealing with critiques of their elephant programs, and California led the charge against their SeaWorld Park more vigorously than, say, Texas did. At the same time, the left is also typically more strongly associated with support for conservation and education programs, which has led to plenty of collaborations with zoos.
In the end, I think zoos defy easy political associations. Newt Gingrich had said that if he hadn't gone into politics, he'd have wanted to be a zookeeper (and even wrote a forward to a book on zoos). I've heard Chelsea Clinton speak at a book signing about how visits to the Little Rock Zoo as a kid with her parents (Bill and Hillary) inspired her love of animals, and she was the keynote speaker at an AZA conference back in 2019 (where she was signing a children's book about wildlife conservation that she wrote). Dan Ashe, current President of AZA, headed USFWS under the Obama Administration. There is a bipartisan Zoo and Aquarium caucus on Capitol Hill, and members of both parties turn out every year to come to AZA's Zoo Day in DC. In my experience, the vast majority of zookeepers are left of center. Private, non-AZA zoo owners, in contrast, tend to swing very hard to the right (mostly due to their severe dislike of regulation).
Overall, though, I would say most politicians of either party DON'T give zoos the least bit of thought, either way. If you were to ask Donald Trump, or Joe Biden, or JD Vance, or Kamala Harris, point blank what their position on zoos was, I think they'd probably stare blankly for a second while they wracked their brains, probably say something very generic and noncommittal, and then move on.