PETA's basic argument is this: "Jesus was compassionate, but slaughterhouses are cruel to animals. Thus, Jesus must have been a vegetarian. He opposed the animal sacrifices of the Jews, and he especially had compassion for fish, calling multiple fishermen away from their occupation of killing innocent animals."
What does Scripture actually say? The Old Testament describes numerous animal offerings commanded by God, including the Passover. Since Jesus was born under the Law, he participated in the Passover meal, which required a lamb to be slaughtered and consumed (Ex 12). The Last Supper was one such meal (Lk 22:8, 15). But this was not Jesus' first taste of lamb, since he had celebrated Passover annually since childhood (Lk 2:41). If Jesus opposed the slaughter and consumption of animals, the New Testament tells nothing of it. Paul even warns against proposing such a doctrine of demons, requiring others to abstain from God-given food (1 Tm 4:3).
Did Jesus call "multiple fishermen away from their occupation of killing animals?" Twice Jesus led disciples to a catch so large that the nets nearly broke (Lk 5:2-7, John 21:5-11). The risen Lord ate fish in front of the disciples (Lk 24:42, 34), guided them in another massive catch, and had them bring the fish to eat with him (Jn 21:9-12).
While it is clear that Jesus ate meat and did not condemn others for doing so, Christians must still treat animals with kindness. "God entrusted animals to the stewardship of those whom he created. . . . Hence, it is legitimate to use animals for food. . . . It is contrary to human dignity to cause animals to suffer or die needlessly . . . [but] one should not direct to them the affection due only to persons" (CCC 2417, 2418).