There is no biblical or religious evidence that Pets/ animals go to heaven when they die. According to Isaiah 11:6 and Isaiah 65: 25, there will be animals in heaven but we do not know if it would be the same animals that died here on earth that will be be raised from the died in heaven.
Pets/ animals have souls, it is believed that everything that has breath and blood has a soul. God created animals and breath of life resides within them. I think their souls do not continue in existence after they die. I do not believe that God created diversity of animals just to live on earth ,be source of food, be part of history and not part of eternity. I strongly believe there is a place for animals in eternity.