Around 50% of all Czech zoos (both big municipal ones and small private ones) allow visitors to bring their pet dogs in.
Other types of pets are usually prohibited. Some zoos have their own free roaming barn cats but those are "working" animals, not pets of visitors.
BTW, this weird American trend of owning a pet miniature horse or a tame parrot, calling them a service/support animal and then demanding entry to places where animals are prohibited is weird and would not fly around here.
Most of zoos that allow dogs in demand you to buy a dog ticket. Prague dog ticket costs currently
$6,50 and you can also buy a dog annual pass for $21. You get free dog waste bags and the zoo provides bowls with fresh water at several places within its area.
The only large local zoo I know that allows dogs in for free is Zoopark Chomutov. You will see a lot dogs inside because its wooded areal of 112ha (277 acres) is just ideal for long walks.
During times of previous Prague zoo director Mr. Fejk, zoo employees (including keepers) were allowed to bring their dogs to their workplace. The new director probably discontinued it.
Brno zoo took a special aproach and attracts dog owners that want to train their pets - recently it even built a dog agility training playground that zoo visitors are free to use during their zoo visit.
Here is small video from Brno with obedience training