Incorrect. PETA is a huge advocate for animal adoption, many members already have pets.
The problem, I'm guessing, is one of language. In PETA, "pet" is a four-letter word. You'll see this occasionally, the word "pet" is used in scare quotes, accompanied with descriptions of how people basically treat "pets" as no different from furniture. There was a case where a woman
forced her son to kill his hamster with a hammer because he got bad grades, another instance where a man
killed his brother's cat to make a point after an argument, people on the PETA2 boards were understandably upset.
They are very fond of the concept that animals should be treated as
companion animal, not
pets (a word they view as derogatory). They really dislike the idea of
buying a living breathing animal as if it were property, they prefer
adopting instead. People should not think of themselves as animal
owners, but
caretakers and
stewards. They avoid the inanimate pronoun
it in favor of
he or
she. See
pro-animal pronouns.
The "institute of pet ownership" refers to a practice of breeding animals for profit. PETA really really dislikes this practice, they view breed-for-profit industries as directly causing a huge amount of cruelty to animals and creating a culture where people disregard animal well-being as if they were furniture.
To an outsider, there's no distinction between a "pet" and a "companion animal", which leads to misunderstandings when PETA
rails against pet shops and breeders but simultaneously
recommends adopting homeless animals from shelters instead. PETA activists are perfectly happy to spell it out for you, like I've just done, if you just take the time to
ask rather than
guess what their policies are regarding pet ownership.
(I personally don't make the pet/companion animal distinction very often when I talk to people, because it makes me seem more pretentious than I really am.)