That said, this reminded me of the response the widely respected Catholic theologian Fr. John Hardon, SJ gave some time ago. In The Catholic Faith magazine (May/June 1999 ~ Vol. 5, No. 3), Fr. Hardon was asked the same question. His reply touches on aspects unaddressed by Dennis:
Pets, as pets, do not go to Heaven. But animals and such like beings may be said to be brought to Heaven because, after the Last Day, they can serve as part of the joys of Heaven. In other words, animals and such like creatures may be said to be brought to Heaven to serve as part of our Heavenly joys. Clearly, we do not need pets to provide happiness in Heaven. But pets and such like creatures will be brought to Heaven to become part of our creaturely happiness in the Heavenly kingdom. Consequently, we may say that animals and such like creatures may be brought to Heaven by God to enable us to enjoy them as part of our creaturely happiness in Heavenly beatitude.