Are you looking for a scale that specifically measures empathy towards pets?
To my knowledge, the closest scales are "Pat attitude scales", "Animal Empathy Scale" and "Attitudes towards Animal Scale." Please note that they are not exclusively about "empathy" towards "pet" but designed to measure either empathy towards non-human animals or general attitudes towards pets. It is worth mentioning, however, that responses to these scales could be predicted by other empathy-related measures (e.g., Empathic concern subscale of Interpersonal Reactivity Index). It seems that empathy towards animals and humans share some core elements, but do not overlap completely.
Please check out the following articles:
Paul, E. S. (2000). Empathy with animals and with humans: Are they linked?. Anthrozoös, 13(4), 194-202.
Taylor, N., & Signal, T. D. (2005). Empathy and attitudes to animals. Anthrozoös, 18(1), 18-27.