What is that situation, anyway:
Annual/Traditional activity gone awry?
Publicity stunt gone awry?
Individual people's oversights leading to individual people's pets getting out of hand?
One way it's the arranging institutions responsibility, the other it's the shop's, the last it's a truly distributive responsibility.
I would say, "all of the above"....
Per strip 978 it is an annual event - the Feast Day of St. Assisi, with reminiscence by Libby of past events. Presumably organized by the Catholics, but apparently an "all are welcome" social occasion regardless of the event. We are not told if the venue is property of a church, or some other entity(ies) that it is rented from, or donated (good deal for area businesses, as it's cheap PR, and brings in extra traffic (customers))
Libby and Anton want to take advantage of the event for publicity - no evidence they asked the organizers for permission, but no evidence they didn't either. Would have been nice if they did, but presumably not required. It isn't just one-sided in any case, as they are making the parade a better show, for whatever that's worth.... Per strip 981, other businesses also show off their animals in the parade.
It isn't clear if the Crow was part of Libby's stock, or not, (would be worse for her if it was since she is responsible for her critters) but my working assumption is that it wasn't; the person it seemed to have escaped from didn't look familiar and she wasn't in fancy costume like the other FF animal handlers. Jeff the bobcat was NOT part of her stock, he owns Sv. Katz

(per strip 979) It appears in strip 983 that at least some of Libby's stock, particularly the elephant she was riding, did NOT participate in the stampede, and that many of the privately owned pets DID.
I would agree the stampede as drawn did seem a bit odd - most of the animals portrayed as fleeing were predators - less likely to be scared, and more likely to want to check out the action to see if there are any leftover snack-bits laying around... Also many of the shown animals are big - again less likely to be scared... OTOH a stampede of pet mice just isn't impressive (not to mention less likely to be destructive...) call it "artistic license"

I can't see any signs of negligence on the part of the organizers / venue owners. Yes, they put on the event, but that shouldn't be grounds for liability
I also don't see any signs of negligence on the part of Libby and friends - they had no reason to expect the stampede situation, and appeared to have adequate handlers to deal with the expected normal sequence of events...
With the exception of the crow's owner, I don't see all that much evidence of negligence on the other pet owners either. Even the crow lady is marginal, since she had no reason to expect the crow to attack Jeff (or any of the other felines in the parade) as opposed to flying off in search of the nearest corn field...
Tough situation to deal with...
ex-Gooserider