Does Roman Catholicism oppose Pet cemeteries and related practices?

The Church teaches that animals do not have souls. This is what sets us apart from animals. "Then God said, "Let us make man in our image, after our likeness; and let them have dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps upon the earth." (Genesis 1:26). God did not make animals in His image. The Bible also says that God "breathed into [man's] nostrils the breath of life" (Genesis 2:7).

We are allowed to bury our pets. However, we should not take it too far. "One can love animals; one should not direct to them the affection due only to persons. " ~Catechism of the Catholic Church paragraph #2418