You've fallen for a smear campaign from a group called "Center for Organizational Research and Education" formerly known as "Center for Consumer Freedom". You can look up their other campaigns here and form an opinion about from where they get their money and which groups they target:

https://www.sourcewatch.org/index.php/Center_for_Consumer_Freedom

This is specifically on the PETA case:

https://crescentcitytimes.com/richard-berman-the-slander-campaign-against-peta/

Yes PETA helped euthanize pets. But so do many many other shelters around the globe every day as they're overflowing from strays and pets from irresponsible owners and scummy breeders. PETA has a video on their website showing that they did it for cases that could not realistically be rescued. Lifelong neglected pets with very little chances of surviving and multiple diseases. Also a reminder, most of us are responsible for killing animals daily. But it's ok as those are bred for it.

Peta is doing what has to be done. The animals are dumped on them as the last resort. They don't have funds to take care of them all. They can't magically make Americans take care of the pets.