I think the best example of how the people feel about animals can be found in the story's opening paragraphs. Rikki-tikki has just been washed on their property by a flood. He is so weakened and dazed by the ordeal that the little boy believes Rikki-tikki is dead, as he says,
Here's a dead mongoose. Let's have a funeral.
At that point, the mother and father both prove they care for animals in general. The mother suggests that instead of getting rid of Rikki-tikki, they should try to nurse him back to health. They successfully do this, and Rikki-tikki begins making friends with the family.
A bit later in the story, the family proves that they not only like animals, but that they are also knowledgeable about animals. The mother and father witness Rikki-tikki sleeping with their baby child, and the father comments that the child is safer now than before.
The father proves that he cares enough about animals to invest time in understanding their basic behavior patterns and their role in ecological food webs. He knows poisonous snakes live in his area, and he knows having a mongoose around the house is a good natural deterrent against those snakes.