I can understand your sadness; I grew up on a farm surrounded by animals, and we have always had pets around our house (mainly dogs and cats). Each has had its own personality, and we always felt sorrow when one of them died.
Pets were common in Bible times. The prophet Nathan used a story about the love of a poor man for his pet lamb to rebuke David for his sin with Bathsheba (see 2 Samuel 12:1-13). On one occasion Jesus, testing the sincerity of one woman's faith, said to her, "It is not right to take the children's bread and toss it to their dogs" (Matthew 15:26). Unlike their pagan neighbors, however, the Jewish people refused to worship animals.
Don't let your sister's failure to understand your sadness cause a disruption between you. Tell her that you realize she may not be able to understand your feelings since she has never had pets herself, but you hope she will respect your concerns. The Bible says, "Let us behave decently ... not in dissension and jealousy" (Romans 13:13).
At the same time, ask God to help you look beyond your heartache and draw you closer to Himself. Someday this life will be over–but are you prepared for that day? You can be, by giving your life to Christ and seeking to follow Him every day.